Arsenal now have 47 points, two ahead of United, who lost 2-1 to West Ham earlier on the same day.
Arsenal striker Nicklas Bendtner, who started in stead of Emmanuel Adebayor, was sent off after collecting two yellow cards and Everton's Mikel Arteta received a straight red card for elbowing Cesc Fabregas.
The Gunners went to the dressing room trailing 1-0 as Tim Cahill sent a loose ball home after Nicklas Bendtner failed to clear a corner-kick.
But Arsenal restored their flowing game in the second period and proved too strong for the home side.
Eduardo equalized in the 47th minute after controlling a long ball from Gael Clichy before sliding the ball past Tim Howard.
He completed his first Premiership double 11 minutes later with a similar effort, taming another pass from Clichy, rounding his marker and slamming the ball home.
Substitute Adebayor capitalized on a howler from Howard, who was too slow to clear the ball, allowing the lanky striker to nip between him and Joleon Lescott before finishing in an empty net.
Tomas Rosicky put the icing on the cake in the 90th minute when he capped a string of passes between Adebayor and Abou Diaby with a low shot for the fourth goal.