Joe Cole - who re-joined West Ham from Liverpool on Friday - had looked set to inspire the Hammers to victory in a fairytale second debut.
The midfielder created both headed goals for the excellent James Collins either side of half-time to cancel out Tom Cleverley's opener.
However, substitute Van Persie stepped off the bench to score a wonderful equaliser in the last minute of injury time to set up a replay at Old Trafford on 16th January.
It had been a bright start by Manchester United and the visitors thought they had gone ahead on 16 minutes when Danny Welbeck swept home a low cross from Javier Hernandez but the offside flag was up.
On 23 minutes, Sir Alex Ferguson's side took the lead, as another Hernandez cross was met by Cleverley and the midfielder swept the ball home in acres of space at the back post.
However, just four minutes later, West Ham were level when Cole had the ball on the left and his floated centre was met with a powerful header by centre-back Collins, which left United goalkeeper David De Gea rooted to the spot.
West Ham turned the tie around on the hour mark, with Cole again the provider, as he delivered another perfectly-weighted ball from the left which Collins met with a header.
With 23 minutes left, Ferguson had seen enough, making a double change as Van Persie and Antonio Valencia replaced Hernandez and Paul Scholes.
On 77 minutes, new boy Cole was replaced by Matthew Taylor, leaving to a standing ovation from the home support.
It was the away fans, however, who were cheering in the final minute as Van Persie broke into the right side of the penalty area on to a stunning cross-field pass from fellow substitute Ryan Giggs, beautifully controlled the ball, held off James Tomkins and expertly drilled the ball into the far corner.