I thought it'd be fun to actually write out what would happen if the Justice League and the Avengers battled each other.
For the sake of simplicity, let's go with a smaller five-man team for the League, and the six-man team of Avengers from the movie. The number difference shouldn't matter.
The Justice League
Green Lantern (Hal Jordan)
As the two teams meet in the middle of a deserted street, Superman and Captain America step forward and trade words. Each leader explains his moral stance on whatever issue they're fighting over, and warns that he and his team won't allow the other to continue in whatever they're doing. A final word is exchanged, then both teams ready themselves for battle.
Black Widow sits quietly in a clocktower, sniper rifle in hand. She takes aim at Batman's head, but is surprised when he turns, looks directly at her, then grins. He taps the bottom of his fist, and Natasha flips her gun over to see a bat-tracer attached to the bottom of the grip. Batman has been tracking her for hours. He talks to someone over his earpiece, and the Flash suddenly appears from out of nowhere at Natasha's side, knocking her out with a super-speed punch.
Captain America rushes forward to attack Batman. The two are evenly matched. Captain America is the literal peak of human ability, and Batman has mastered nearly every form of martial arts known to man. They dodge, weave, and strike at each other, neither one able to land a strong blow. Every gadget that Batman throws, Cap is able to block with his shield, and Batman is too smart and skilled to be knocked out by the shield.
At the Justice League watchtower, sometime before the battle, Batman lays out the League's plan of attack. Of particular note is the recommendation that Superman stay away from Thor, since magic, like kryptonite, is one of Superman's weaknesses.
Meanwhile, at Stark Tower, Captain America recommends that Thor tackle Superman directly, since Superman should be weak against Thor's lightning.
Back in the present, Thor rushes to engage Superman, but is quickly blocked by Wonder Woman. Thor looks to the Hulk for help, but Hulk is too enraged to be much of a team player, instead leaping fists-forward at Superman.
Hawkeye sits atop a nearby building, ready to snipe the Leaguers with his arsenal of trick arrows. As the Flash exits the building where he knocked out Black Widow, Hawkeye—with his extremely high level of perception—manages to predict Flash's movements, nocking an explosive arrow and lining up the shot.
Iron Man fires a few repulsor blasts that Hal easily blocks with energy shields. Hal fires back a few energy blasts that Iron Man dodges or is able to take without stumbling. Tony launches a cluster of mini-missiles; Hal creates a mini-gun energy construct and obliterates the missiles in mid-flight with a hail of bullets. Tony gets tired of being blocked and moves in close. Hal's energy aura protects him to a degree, but Tony's punches are actually able to hurt a bit. Hal and Tony wrestle in the air for a while, but eventually Tony gets impatient and uses his armor's chest beam to unleash an insane level of energy at Hal. Hal stumbles back, but then counters with a giant beam of his own. At this point, it's the power of Tony's arc reactor against Hal's power ring. Theoretically, both devices would run out of power roughly at the same time, but the power ring has a higher level of immediate output by an astronomical factor, limited only by the wielders' willpower. Hal overwhelms Tony in an explosion of light, blasting parts of the Iron Man armor clean off and damaging the rest of the suit beyond repair. Tony falls to the ground, unconscious. Hal, exhausted, floats slowly to the ground. A voice echoes from the ring: "Ring charge at 2%."
Hawkeye fires the explosive arrow at Flash; Flash barely manages to dodge it. Hawkeye quickly fires off a series of explosive arrows, each one singing Flash's suit. Flash eventually manages to zig-zag his way closer and closer, finally running and taking a long jump towards Hawkeye. Clint smirks; Flash can't dodge an arrow if he's suspended in mid-air. As Clint fires the arrow, Flash vibrates his body at super-speed, passing right through the arrow unharmed. Before his feet touch the ground, Flash's fist hits Hawkeye's face. Instant knockout.
Flash turns to go help Batman with Captain America. Captain America, however, with his perfectly strategic mind, is able to calculate Flash's exact angle and velocity. As Flash nears the duo, Cap swings his shield, backhanding Flash in the face with a wall of solid vibranium. Flash hits the floor, out cold.
As Thor and Wonder Woman fight, Thor is shocked to discover that Wonder Woman's bracelets easily block lightning. Wonder Woman tries to restrain Thor with the lasso of truth, but Thor grabs hold of the lasso and yanks it, pulling Diana close and slamming her in the face with his knee. Diana recoils from the hit, but isn't injured. As the battle goes on, neither gets the upper hand. Thor is stronger, but Wonder Woman is far more skilled. If Thor could land a full-force blow with Mjolnir he'd win instantly, but Diana is simply too fast. On the other hand, Thor is so tough that Diana can't seem to land an effective hit.
The Hulk grabs Superman by the cape and swings him directly into Wonder Woman, sending them both tumbling across the street. As they both get up, they find themselves facing different opponents. Superman and Thor stare each other down while Wonder Woman prepares herself to fight the Hulk.
Wonder Woman unsheathes the sword at her belt. Tipped with magic, it's something she'd never use against a human for fear of killing him, but against the Hulk she'll need all the help she can get. The Hulk swings at her wildly, but she dodges him and jabs her sword at Hulk's knees and shoulders. The Hulk's skin isn't invulnerable, especially not against magic weapons, and he begins to slump down to the ground.
Green Lantern decides to use the remaining 2% of his power to help Batman defeat Captain America. He fires off his remaining power in one last blast... which Captain America not only blocks, but deflects directly into Wonder Woman's side. Wonder Woman isn't seriously hurt, but the break in her attack is enough for the Hulk to fight back. He pummels her repeatedly until she lay unconscious in a pile of rubble. At the same time, Batman is able to take advantage of Cap's distraction and throws a freeze grenade. Cap is instantly frozen in a large chunk of ice—something Batman finds pleasantly ironic.
Thor knocks Superman back for a moment, then summons an immense bolt of lightning and unleashes it directly into Superman. Superman yells in agony for a moment, but as the lightning fades he still stands. Both Thor and Superman are momentarily shocked, but Superman quickly realizes the truth: Thor's lightning is the same as any other lightning on Earth; it's elemental, not magical. Superman rushes forward, throwing a flurry of punches at Thor. Thor kicks him back, then throws Mjolnir. Superman barely ducks under Mjolnir's path, then leans over to pick up the hammer from the ground. Thor almost chuckles; only the worthy can lift Mjolnir. Superman is more than worthy. He lifts the hammer and slams it into Thor's jaw. Thor is out.
Green Lantern is powerless and Batman is entirely useless in the fight. Superman and Hulk are all that's left. They trade blows, each hit sending out shockwaves that shatter windows and crack stone walls across the city. Superman can't seem to do real damage against the Hulk with his fists; he fires a heat vision blast into Hulk's eyes. Hulk screams in pain and steps back, then opens his now-reddened eyes and roars with anger. Hulk slams Superman with punches that would level entire buildings. Superman coughs up blood, then wipes it from his mouth and decides enough is enough. He flies behind the Hulk, lifts him off the ground, and flies into space. Superman is capable of surviving in space for hours at a time, so long as he's near a yellow sun. The Hulk, however, still needs to breathe. The Hulk gasps for air that doesn't exist, then falls unconscious.
Back in the city, Batman and Green Lantern jump back, startled, when the Hulk falls from the sky and slams into the street below.
Justice League wins.