The Blues looked on course for the three points after Didier Drogba headed them into the lead with 52 minutes played.
However, United salvaged a share of the spoils when Van Persie smashed the ball in from close range with 60 seconds of added time remaining.
Here, Sports Mole takes a closer look at the action between the two teams at Old Trafford.
On target: 7
On target: 4
Was the result fair?
Despite leaving it so late to draw level, it is hard to argue against Manchester United deserving their share of the spoils. Both teams took part in a thrilling contest that ebbed and flowed, and it would be right to say that a draw was the fairest outcome.
Man Utd's performance
The Red Devils came flying out of the traps at the start of the match, exploding into life early on to put Chelsea under pressure. Angel di Maria, Juan Mata and Adnan Januzaj all looked dangerous, while Marouane Fellaini impressed in midfield with a robust and disciplined display. There was also plenty of bite up front as Van Persie's movement and unselfish play caused the visitors problems.
Louis van Gaal's side saw their intensity wane after the break, allowing Chelsea to take control of the match for much of the second half, but it was their persistence that shone through in the end as they came away with a decent point against the Premier League leaders.
The Blues had to be organised from a defensive point of view in the opening half as United kept them pinned back in their own half for large periods. Jose Mourinho's men were limited to the occasional counter-attack, while John Terry and his colleagues at the back soaked up a fair amount of pressure.
However, the away side took the game to United after the break and went deservedly ahead through Drogba's goal. They had chances to put the game to bed after that thanks to some incisive attacking football, but they lacked the vital touch to kill their opponents off. Even though they were so close to claiming the win, Chelsea are likely to be relatively satisfied by keeping their unbeaten record intact.
Sports Mole's man of the match
Thibaut Courtois: Chelsea's goalkeeper was superb between the sticks, keeping United out on numerous occasions with some fine saves. His personal battle with Van Persie was intriguing as he foiled the Dutchman three times before he was finally beaten late on.
Rafael did a very Rafael thing with just 12 minutes played when he was cautioned for pulling back Eden Hazard. The Brazilian had to be careful for the remainder of the match, putting his side in jeopardy, but luckily for him he remained out of trouble.
Phil Dowd was kept busy during what was a physical match between the sides. He showed little fear in brandishing his yellow card, while the decision to send off Branislav Ivanovic late on was the correct one. The official could have awarded Chelsea a penalty in the first half when Ivanovic was hauled down in the box leading up to a corner, but it would have been a controversial moment had he done so.
Man Utd: United continue their Premier League campaign next Sunday when they make the short trip to local rivals Manchester City.
Chelsea: The Blues are in League Cup action on Tuesday away to League Two side Shrewsbury Town.