Tom Cleverley gave the visitors a 23rd-minute lead at Upton Park, but Joe Cole, making his first West Ham appearance since his return from Liverpool, twice set up James Collins to give the hosts a 2-1 lead.
Alex Ferguson's side were drifting towards a second third-round exit in three seasons, after a 1-0 loss to Leeds United in 2010, until van Persie finished from Ryan Giggs' raking pass to take the tie to a replay.
Earlier, debutant Demba Ba scored twice as holders Chelsea roared back from behind to beat Southampton 5-1.
Ba, who signed from Newcastle on Friday, cancelled out Jay Rodriguez's 22nd-minute opener and then added a second as Chelsea ran riot, with Victor Moses, Branislav Ivanovic and substitute Frank Lampard also on target.
"His movement and understanding of things we were wanting to do was really good. He was very clever and he could understand the movement of the players. He did everything well on the pitch," said Chelsea interim manager Rafael Benitez.
English champions Manchester City downed second-tier Watford, with Carlos Tevez scoring a ferocious free-kick before Gareth Barry and Marcos Lopez completed a 3-0 success.
The third round had opened with an upset as second-tier Brighton and Hove Albion beat Newcastle 2-0 to knock Alan Pardew's side out of the competition for the second year in succession.
Andrea Orlandi and Will Hoskins scored in either half at the AMEX Stadium, while Newcastle, who fielded a weakened team, had captain Shola Ameobi sent off for two bookable offences.
"If I am being honest, that team is a shadow of the team we put out last season," said Newcastle manager Pardew.
"We need to get some bigger players back, who can handle the situation of carrying the Newcastle jersey."
The biggest upsets of the day saw Macclesfield Town and Luton Town of the fifth-tier Conference National overcome Championship sides Cardiff City and Wolverhampton Wanderers, 2-1 and 1-0 respectively.
Premier League sides Fulham, Sunderland, Wigan Athletic and Stoke City were all taken to replays by lower-league opponents.
Jordi Gomez levelled for Wigan in a 1-1 draw against third-tier Bournemouth, while Sunderland hit back from two goals down to draw 2-2 at Championship representatives Bolton Wanderers.
Giorgios Karagounis' stunning 80th-minute volley rescued a 1-1 draw for Fulham at home to Blackpool, and Stoke City were held to a 0-0 draw by Crystal Palace despite giving a first start to close-season signing Michael Owen.
Aston Villa fought back at home to Ipswich Town to prevail 2-1 through goals from Ipswich old boy Darren Bent and Andreas Weimann, while Reading also had to come from behind in a 3-1 win at Crawley Town.
Tottenham Hotspur avoided a repeat of their 1987 cup final loss to Coventry City with a 3-0 success at White Hart Lane that saw Clint Dempsey claim a first-half brace.
In one of four all-Premier League encounters in the third round, Kieron Dyer's injury-time effort earned QPR a 1-1 draw at home to West Bromwich Albion.
Seventh-tier Hastings United saw their campaign ended at Championship club Middlesbrough, who prevailed 4-1 despite seeing on-loan Hastings goalkeeper Liam O'Brien save an 18th-minute penalty from Ishmael Miller.
Meanwhile, Stale Solbakken was sacked as manager of Championship side Wolves on Saturday, just hours after the club had been dumped out of the FA Cup by non-league Luton.
The struggling club, who were relegated from the Premier League last season, also terminated the contracts of assistant manager Johan Lange and Patrick Weiser, the first team coach.
Wolves are currently 18th in the Championship and have won just three out of the last 16 league games and a disappointing run of results culminated in Saturday's FA Cup third round 1-0 defeat to Luton Town.
Solbakken had only been in charge at Molineux for six months.
"Kevin Thelwell, head of football development and recruitment, will take charge of first team training until a new manager is appointed, assisted by development coach, Steve Weaver," said a club statement.
"The club would like to offer their thanks and best wishes to Stale, Johan and Patrick."