Joe Cole had delivered a vintage display on his return to his boyhood club on a free transfer from Liverpool this week, and rolled back the years as he set up a double for defender James Collins with superb crosses from the left flank in each half to turn the tie around after Tom Cleverley had fired United into the lead on 23 minutes.
However, as the clocked ticked into stoppage time, Dutchman Van Persie, who had started on the bench, ghosted into the right side of the penalty area, before drilling a low shot into the bottom corner to force a replay.
Danny Welbeck and Javier Hernandez had opened up in attack for the Barclays Premier League leaders, who also had Nemanja Vidic and Paul Scholes in the side, with Rio Ferdinand, Antonio Valencia, Ryan Giggs and Ashley Young all joining Van Persie on the bench in reserve.
It was a bright start by United as Cleverley tried to release Hernandez down the left, but the ball rolled through to West Ham keeper Jussi Jaaskelainen.
The hosts were cut open again on nine minutes when Chris Smalling got clear on the right and cut a low pass through the penalty area, which was just in front of Hernandez.
West Ham suddenly burst into life when Guy Demel darted down the right and his cross was put behind.
From the resulting corner, Alou Diarra's downward header was nodded off the line by Scholes and Carlton Cole could not put the rebound away as Vidic cleared.
Captain Kevin Nolan, who missed the Barclays Premier League win here over Norwich on New Year's Day through suspension, then just failed to get on the end of a flick into the six-yard box from Carlton Cole.
United thought they had gone ahead on 16 minutes when Danny Welbeck swept home a low cross from Hernandez, but the offside flag was up against the Mexican.
West Ham defender Daniel Potts then hacked a goalbound effort from Vidic off the line after a corner caused more mayhem in the hosts' penalty area.
It was 1-0 to United on 23 minutes.
More slick play down the right from Rafael released Hernandez. His low centre went all the way across into the path of the advancing Cleverley, who swept a brilliant first-time effort back past the keeper into the bottom corner.
The Irons immediately went on the offensive.
Joe Cole's free-kick picked out Ricard Vaz Te, but his tame header was straight at the keeper.
Collins showed him how it should be done on 27 minutes as the home side levelled.
The ball was pushed out to Joe Cole on the left, and his floated centre was met with a powerful header by centre-back Collins, which left United keeper David De Gea rooted to the spot as the ball flew into the net.
Even though they soaked up plenty of pressure, the visitors remained dangerous on the break.
Welbeck sprinted away down the left and on into the West Ham penalty area before unleashing a fierce angled drive which Jaaskelainen beat away.
At the start of the second half, Cleverley was found in space on the edge of the area, but shot wildly over the bar.
Joe Cole refused to give up on a loose ball at the right side of the penalty area, hooking it back from the goalline and into the six-yard box, where Carlton Cole glanced a header towards the far corner which De Gea collected at full stretch.
West Ham turned the tie around on the hour mark.
Joe Cole was again the provider, as he floated over a perfectly-weighted ball from the left which Collins again met with a powerful header into the far corner.
With 23 minutes left, Sir Alex Ferguson had seen enough, making a double change as Van Persie and Valencia replaced Hernandez and Scholes.
On 77 minutes, Joe 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 superb cross-field pass from fellow substitute Ryan Giggs, held off James Tomkins and expertly drilled the ball into the far corner to force a replay back at Old Trafford.