Two headed goals from James Collins had put the Hammers in front but substitute Van Persie intervened with virtually the last kick of the game.
The Dutch striker isn't expected to start for hosting United on Wednesday though, with the Red Devils likely to hand a starting role to Mexican international Javier Hernandez.
The last time these two teams met in the FA Cup at Old Trafford, Man United ran out 6-0 winners in 2003.
Man United: WDWWWW
West Ham: LDLLWL
Possible starting lineups:
Man United: Lindegaard; Buttner, Smalling, Jones, Rafael; Valencia, Scholes, Anderson, Nani; Rooney, Hernandez
West Ham: Jaaskelainen; Reid, Demel, Potts, Collins; Diarra, Collison, Nolan, Jarvis; J Cole; Chamakh
Sports Mole says: 3-1