United has enough points to ensure a record of only one win in three matches hasn't made an impact in the standings.
Ricardo Vaz Te put West Ham in front in the 16th minute, completing a counterattack that Mohamed Diame started by charging with the ball from inside his own half before releasing Matt Jarvis on the left flank.
After reaching the byline, Jarvis centered the ball for Carroll to connect with a downward header and Vaz Te to find the final touch, nodding into the net from close range.
But United leveled 15 minutes later after Shinji Kagawa worked his way down the left flank and knocked the ball back to Antonio Valencia, who had the simple task of tapping in his first goal for a year.
West Ham regained the lead in the 55th when Vaz Te flicked the ball to Guy Demel, whose pass allowed Diame to turn and lose his marker before curling the ball into the net.
Van Persie came close twice before pulling United level again in the 77th, sending the rebound into the net after Kagawa's strike came off both posts.
West Ham manager Sam Allardyce thought Van Persie was offside.
"Andy Carroll has a shot two minutes in and his flag went straight up, so I don't know what's going on," Allardyce said. "This has taken a famous victory away from us today. To draw this game by a default goal from Manchester United is difficult to swallow.
"A fantastic performance by the team they get rubbished for some of the football they play, and that's a disgrace as well."
In Manchester, City's only goal came when Carlos Tevez weaved his way into the penalty area and unleashed a shot into the roof of the net in the 83rd.
The victory left Wigan City's opponents in the FA Cup final next month three points from safety having played a game fewer than Aston Villa.
At Craven Cottage, Terry scored twice but the opener from fellow defender David Luiz eclipsed it for quality.
Eden Hazard squared to Luiz, who took a touch before sending the ball swerving into the net on the half-hour from 35 meters.
Terry headed in his first before halftime from Juan Mata's cross, and the captain found the net again in the 71st, helping Fernando Torres' header from Mata's corner find the target.
- AP