United now has 70 points from 30 games, Arsenal 67 and Chelsea, which led 3-1 and squandered the lead three times at White Hart Lane, has 65. Fourth-place Liverpool appears to be out of contention with 59.
Chelsea manager Avram Grant rued the fact that his team led 1-0, 3-1 and 4-3 and still couldn't beat Spurs.
"When you lead three times and you don't win the the game, I'm not so happy with this," he said. "It's good for the supporters but we wanted to win and we didn't so it's disappointing. The team are not so happy."
Didier Drogba headed home a cross from John Terry to give Chelsea a third-minute lead at White Hart Lane. But Spurs, who beat the Blues 2-1 in last month's League Cup final, leveled in the 12th when Jermaine Jenas swung in a free kick from the right and Jonathan Woodgate scored with a powerful header.
Chelsea regained the lead in the 20th minute when Joe Cole dribbled through the Spurs defense and supplied a pass to Michael Essien to chip the ball over goalkeeper Paul Robinson who had raced off his line.
Cole made it 3-1 in the 52nd after taking a pass from Claude Makelele, but Dimitar Berbatov replied for the Spurs in the 61st, beating Terry in the air and heading home a cross from Tom Huddlestone.
Huddlestone then equalized for 3-3 in the 75th minute, but Cole edged Chelsea back in front five minutes later.
It appeared that Chelsea would draw level on points with Arsenal but Keane struck again with a shot into the top corner in the 88th minute to make it 4-4.
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