Senegal international Ba, signed from Newcastle United on Friday, scored the London side's equaliser and their fourth before Frank Lampard's late penalty took the midfielder to joint second on Chelsea's all-time scoring list with 193 goals.
United, 11-times FA Cup winners but not since 2004, were staring at a 2-1 defeat at Upton Park after Joe Cole marked his return to East London by setting up two carbon-copy headers for James Collins after Tom Cleverley put United ahead.
With 90 minutes on the clock, substitute Van Persie rescued his side by controlling a 40-metre pass from Ryan Giggs and dispatching a lethal shot past keeper Jussi Jaaskelainen.
"You never rule out fighting back with our team but certainly at 2-1 down with a couple of minutes to go you hope you get a break," United manager Sir Alex Ferguson said.
"The manner of the goal, with Ryan Giggs's ball and Robin van Persie's finish, was world class."
While Ba began in style, the gloom remained at his old club Newcastle, who lost 2-0 at Championship Brighton & Hove Albion - one of several upsets on the day when England's lower-ranked clubs traditionally take aim at the big boys.
Goals from Andrea Orlandi and Will Hoskins earned Brighton, runners-up to Manchester United 30 years ago, a repeat of their fourth-round victory over Premier League Newcastle last season, condemning Alan Pardew's side to an 11th defeat in 14 matches.
Cardiff City, runaway leaders atop the Championship, were the biggest victims, losing 2-1 at non-league Macclesfield Town, while former FA Cup runners-up Luton Town, now outside the Football League, beat visiting Wolverhampton Wanderers 1-0.
Several clubs struggling in the Premier League flirted with defeats against lower league opposition before earning replays.
Fulham were held 1-1 at home by second division Blackpool, Wigan Athletic needed a late equaliser in a 1-1 draw with third-tier Bournemouth, and Sunderland recovered for a 2-2 scoreline at Bolton Wanderers, who lost their top-flight status last season.
Premier League Reading had to come from a goal down after one minute to beat third-tier Crawley Town 3-1 away.
South coast seventh-tier minnows Hastings United, the lowliest survivors of the 758 clubs who began the 132nd edition of the world's oldest knockout cup competition in August, had their fairytale cut short in a 4-1 defeat at Middlesbrough.
Chelsea's Ba needed only 35 minutes to start justifying the seven million pounds fee British media reported Chelsea had paid for his services when he crashed the ball into the net from a position virtually on the goal-line after Juan Mata's chip.
Jay Rodriquez had given Southampton an early lead but once Ba levelled they were no match for Rafa Benitez's side. Victor Moses made it 2-1 just before half-time and Branislav
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