Manchester United's seven-point lead over neighbor Manchester City stayed intact after the title-chasing rivals downed two of the Premier League's fallen giants in contrasting ways on Sunday.

While United scraped to a nervy 2-1 win at home to Liverpool, City sauntered to a 2-0 victory at Arsenal after playing against 10 men for most of the match.

Laurent Koscielny's ninth-minute red card for wrestling Edin Dzeko to the ground proved decisive, with Dzeko missing the resulting penalty but later adding to the superb opener by James Milner. Vincent Kompany was sent off late on for City.

Robin van Persie set United on its way to victory at Old Trafford with his league-high 17th goal. Nemanja Vidic made it 2-0 but United relied on some desperate defending to preserve the lead after Daniel Sturridge's goal.

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Manchester City capitalized on Laurent Koscielny's early sending-off to canter to a first league victory at Arsenal in 38 years, with Sunday's 2-0 win keeping the defending champions seven points behind Manchester United in the Premier League title race.

The France center back was shown a straight red card for rugby-tackling Edin Dzeko to the ground inside the area, leaving Arsenal a man down with just nine minutes gone.

Dzeko's resulting penalty was saved — via the post — by Wojciech Szczesny, but the Bosnia striker made amends by adding to James Milner's opener to give City a 2-0 lead after 32 minutes.

The visitors also ended the game with 10 men after captain Vincent Kompany was red-carded for a wild two-footed challenge on Jack Wilshere, even though he got the ball.

City's record at Arsenal was so bad that the champions had to go back to 1975 for their last league win at the home of the London club. They hadn't even scored a goal in their last five league visits to Emirates Stadium.

However, there was only going to be one winner once Koscielny was dismissed for wrestling Dzeko to the ground as they competed for a ball that had been flicked back into the penalty box.

Koscielny's vehement protests at referee Mike Dean were in vain and although Szczesny saved Dzeko's penalty with his feet, with the ball rebounding off the post and back into the goalkeeper's arms, the respite was brief.

Milner, one of manager Roberto Mancini's go-to men in big games, rifled a fine finish in off the post in the 21st minute after being fed by Carlos Tevez's no-look pass, and then helped set up Dzeko's goal.

Tevez saw his flicked shot from Milner's low cross palmed away by Szczesny but only into the path of Dzeko, who tapped into an empty net.

Jack Wilshere cleared Javi Garcia's header off the line in the final moments of the first half as City continued to press and it was more of the same in the second half until Kompany's red card for diving in two-footed toward Wilshere.

It gave Arsenal a fighting chance of salvaging something from the match — and substitute Olivier Giroud should have done better than heading a free kick over from eight yards out — but City held on.

Kompany's three-match ban will cause him to miss Premier League games against Fulham and Queens Park Rangers as well as the FA Cup fourth-round match at Crystal Palace or Stoke. It leaves City deprived of another leading player, with midfielder Yaya Toure at the African Nations Cup potentially until mid-February.

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Manchester United weathered a late fightback fromLiverpool to secure a 2-1 win over its great rival on Sunday, tightening the leaders' grip at the top of the Premier League.

Robin van Persie's league-high 17th goal of the campaign gave United a 19th-minute lead, and the hosts looked likely to coast to victory at Old Trafford when captain Nemanja Vidic unwittingly deflected the ball in from close range to make it 2-0 in the 54th.

Daniel Sturridge pepped up a lethargic Liverpool when he came on as a halftime substitute and the new signing from Chelsea halved the deficit three minutes later, but Brendan Rodgers' team couldn't find the equalizer to cap its improved second-half display.

United is now 10 points clear of second-place Manchester City ahead of the champions' match at Arsenal later Sunday, with its total of 55 points after 22 games the team's highest tally at this stage of a top-flight season.

The game between England's two most successful clubs, which share 37 league titles between them, was billed by some as a duel between Van Persie and Liverpoolstriker Luis Suarez — the division's most prolific marksmen. Van Persie came out on top.

While Suarez was kept quiet — much to the delight of home fans who jeered his every touch as a result of his recent spats with United defender Patrice Evra — his rival continued his sensational scoring run that is keeping City at arm's length in the title race.

The Netherlands striker beat marker Daniel Agger to Evra's driven cross and calmly slotted the ball home from 12 yards for his ninth goal in his last nine games. Van Persie has also scored at least one goal in nine of his last 10 games.

"Robin van Persie is different class," United striker Danny Welbeck said. "He produces moments in the game others can't do and we are glad he is on our side."

With Liverpool's midfield giving the ball away cheaply and sitting far too deep, United was invited to attack at will and should have been more than a goal ahead at halftime.

Welbeck, given a rare start up front with Wayne Rooney still injured, thrashed wildly wide and Tom Cleverley went close with a volley. Van Persie then saw an impudent flick cleared from near the goal-line by Liverpool defender Martin Skrtel before Pepe Reina pushed the ball away from the inrushing Shinji Kagawa.

The first half highlighted the gap that has emerged between the two sides in recent years, with the last of Liverpool's 18 titles coming back in 1990. It was much different after the break, though.

Sturridge was introduced for the second half for his league debut for Liverpool and his pace and energy sparked a vast improvement by the visitors, who finally tested the United defense.

The former Chelsea striker's goal came only after United had stretched its lead, however, when Van Persie's free kick was headed down by the unmarked Evra with the ball deflecting off the side of Vidic's head into the net.

Liverpool's riposte was almost immediate, Sturridge sidefooting home from close range after David De Gea could only palm out Steven Gerrard's low shot, and it set up a tense last half hour for the home side.

Sturridge and fellow substitute Fabio Borini shot wide as United fans became increasingly anxious, and Sturridge had an even better opportunity in the 85th only to blaze over the crossbar from an angle.

United, which has made a habit of edging tight matches this season, is now 24 points clear of eighth-place Liverpool in the standings.