Didier Drogba, 36, playing from the start because Diego Costa was unfit, headed the visitors in front in the 53rd minute from a corner conceded when David De Gea was forced into a fine save from Eden Hazard.
Chelsea seemed likely to hold out but just before the final whistle Branislav Ivanovic was sent off for a second yellow card and from the free kick Thibaut Courtois saved from Marouane Fellaini but could not keep out Van Persie's fierce follow-up.
The draw extended Chelsea's lead to four points over second-placed Southampton and six over champions Manchester City but they would have been expected to see the game out after a strong second half performance.
It was only the second time this season Jose Mourinho's side have failed to take all three points, both games being in Manchester, where they also drew 1-1 with City.
Seven wins and two draws from nine games nevertheless leaves them in an enviable position.
United, meanwhile, having briefly reached fourth place, find themselves back in eighth after a second successive draw.
The consolation for them is having equalised right at the end again, just as they did in a 2-2 draw at West Bromwich Albion last Monday.
Van Persie believes they are steadily improving. "We can be happy with a point," he told Sky Sports. "It's getting better every single day. It's still early doors and we can catch Chelsea, yes."
A year ago Mourinho had been criticised for Chelsea's negative approach at Old Trafford when he played without a recognised striker and achieved a goalless draw.
Without Diego Costa and Loic Remy, there might have been a temptation to adopt the same tactics but starting with Drogba just in front of Willian, Oscar and Eden Hazard reflected the London side's confidence after their impressive start to the season.
"I'm very happy," Mourinho said. "My team played a fantastic game. It was not the result we came for, not so good, but acceptable."
He was reluctant to be drawn on Ivanovic's two yellow cards for fouls on Angel Di Maria but was clearly not happy about the dismissal.
Van Gaal, however, said: "I've seen it on television and he touched him two times."
United's manager added that what pleased him most was the way United kept going, saying: "We showed again that we believe until the end in making a goal."
They face a tough assignment next, in the Manchester derby away to City next Sunday.