United bounced back from defeat at Arsenal before the international break as Everton suffered their first defeat since Aug. 23.
Morgan Schneiderlin opened the scoring on 18 minutes, slotting into the far corner from Chris Smalling's chest-down, and Ander Herrera headed home a second soon afterwards from a Marcos Rojo cross.
David De Gea had to make a good save to deny Romelu Lukaku early in the second half, but United extended their lead with 62 minutes played when Rooney netted against his former club.
The England captain produced a clinical finish from Herrera's throughball for his first away goal in the Premier League since November, and his first at Goodison Park since April 2007, to complete the scoring.
The goal also saw Rooney draw level with former United striker Andy Cole as the second highest scorer in Premier League history.
Chelsea ended their run of three games without a victory as Jose Mourinho turned the spotlight onto his Aston Villa counterpart Tim Sherwood with a 2-0 win at Stamford Bridge.
The Blues went into the game searching for a first win since beating Walsall 4-1 in the Capital One Cup on Sept. 23, and looking to bounce back from their 3-1 defeat to Southampton before the international break.
They were given a significant helping hand for the opening goal on 34 minutes, with Villa keeper Brad Guzan playing a reckless ball out to Joleon Lescott, who failed to control the pass and allowed Willian to burst through before setting up Diego Costa for the simplest of finishes.
They were further boosted at the start of the second half when Alan Hutton deflected the ball into his own net from Diego Costa's pass to give Chelsea a much-needed win.
Raheem Sterling scored a hat trick as Premier League leaders Manchester City eased past Bournemouth 5-1.
City took the early lead against the run of play, with Sterling netting from close range on seven minutes after Wilfried Bony helped on a Kevin De Bruyne knockdown.
Bony then added his own name to the scoresheet four minutes later after Bournemouth keeper Adam Federici had failed to hold a cross, although the Cherries soon reduced the arrears through Glenn Murray.
City restored their two-goal cushion soon afterwards as Sterling added his second of the game, showing superb composure before slotting home, and he had his third just before the break as he sent a shot past Federici at the near post.
City then increased their advantage late on when Bony added his second of the game with a neat turn and finish.
Arsenal had to wait until the second half, but eventually sealed a comfortable 3-0 win at Watford.
Watford had some good chances early on, but Petr Cech was equal to them and Alexis Sanchez broke the deadlock with a strike in the 62nd minute after good work from Mesut Ozil saw the ball fall kindly to him.
Ozil then put the ball on a plate for substitute Olivier Giroud to slot home just five minutes later and Aaron Ramsey's effort made it 3-0 in the 74th minute.
Liverpool played out a goalless draw at Tottenham Hotspur as Jurgen Klopp took charge of the visitors for the first time.
Liverpool made a bright start, and they might well have had the lead when Divock Origi headed against the crossbar.
However, Tottenham battled back, and substitute Clinton Njie -- on for the injured Nacer Chadli -- brought a good save from Simon Mignolet just before the half-hour mark before Harry Kane and Dele Alli went close in quick succession.
The second half brought far fewer chances at both ends, although Kane came close in the 84th minute when his low drive brought a good save from Mignolet, while Emre Can fired just wide shortly before the final whistle.
Leicester City battled back from two goals down to snatch a 2-2 draw at Southampton thanks to two goals from in-form Jamie Vardy.
Leicester, who began the day with 15 points from their opening eight games, fell behind on 21 minutes when Southampton centre-back Jose Fonte headed home.
Fonte's centre-back partner, Virgil van Dijk, doubled their lead before the break, scoring from the follow-up after Graziano Pelle's header had struck the post.
Vardy, the Premier League's leading scorer this season, reduced the deficit on 66 minutes after good work from Nathan Dyer and then produced a fine finish in injury time to secure a share of the spoils.
Sam Allardyce could not bring about an improvement in Sunderland's fortunes as they lost 1-0 at West Bromwich Albion in his first match in charge.
Saido Berahino gave West Brom the lead on 54 minutes, tapping into the net after Sunderland keeper Costel Pantilimon failed to claim a cross from Chris Brunt.
That proved the only goal of the game as Sunderland remain locked in the relegation zone.
West Ham United scored two late goals to see off 10-man Crystal Palace 3-1 at Selhurst Park.
Carl Jenkinson gave West Ham the lead, poking a shot past Wayne Hennessey on 22 minutes, but moments later the Arsenal loanee gave away a penalty for a foul on Dwight Gayle.
Yohan Cabaye restored parity from the spot, despite being ordered to retake the penalty, but the Eagles' hopes of claiming a positive result suffered a setback when Gayle picked up a second booking just before the break.
That proved costly as Manuel Lanzini fired West Ham into the lead in the 88th minute before Dimitri Payet added a third deep into injury time.