A day after completing his move from Newcastle, Ba took just 35 minutes on Saturday to score at Southampton to cancel out Jay Rodriguez's goal.
After further Chelsea goals from Victor Moses and Branislav Ivanovic, Ba netted his second before Frank Lampard wrapped up the rout from the penalty spot.
"It is always important for a striker to score goals but more than just the goals it is the contribution for the team," Chelsea manager Rafa Benitez said.
Newcastle struggled without Ba, however, and was the only English Premier League side to be ousted by lower-league opposition as Alan Pardew's side lost 2-0 at second-tier club Brighton.
At West Ham, Joe Cole had a successful return as he set up two nearly identical goals for James Collins as the Hammers looked to knock off United, but Van Persie came on as a substitute to rescue the draw with a clinical finish in stoppage time.
"The manner of goal — the pass from Ryan Giggs and first touch from Robin van Persie and his finish — is absolutely world class, absolutely fantastic," United manager Alex Ferguson said.
Manchester City had no trouble ousting League Championship club Watford 3-0, while Tottenham beat third-tier club Coventry by the same score.
The biggest upsets were provided by fifth-tier nonleague sides overcoming second-tier opposition as Macclesfield beat League Championship leader Cardiff 2-1, and Luton defeated Wolverhampton Wanderers 1-0.
At Southampton, Ba showed just how valuable he can be for Chelsea's quest for silverware this season as Fernando Torres was rested on the bench.
Ba's 14th goal of the season — and first for Chelsea — came after Juan Mata initially attempted to score from an acute angle, cancelling out Rodriguez's opener.
Chelsea went in front on the stroke of halftime after Moses collected a cutback from Ashley Cole and struck.
Seven minutes into the second half, Ivanovic headed in Mata's cross, and Ba found the target again in the 61st after being setup by Eden Hazard.
The late penalty from Lampard — after Steven Davis handled — allowed the midfielder to move to joint second on Chelsea's all-time leading scorers list alongside Kerry Dixon.
As Southampton went out of the cup, it was a successful day for another south-coast side as Brighton eliminated Newcastle for the second successive season.
Andrea Orlandi put Brighton ahead in the 33rd, hooking in the ball with the outside of his boot after meeting Wayne Bridge's cross.
Injury-depleted Newcastle's prospects took a further turn for the worse when captain Shola Ameobi received a second booking for his late tackle on David Lopez in the 63rd.
Brighton extended its lead in the 87th when Will Hoskins slotted the ball through goalkeeper Rob Elliot's legs.
"We didn't have enough quality players on the pitch today ... the younger players have put as much effort in as they could," said Alan Pardew, whose Newcastle side is just above the Premier League relegation zone.
At Upton Park, United took the lead after 23 minutes when Tom Cleverely met Javier Hernandez's cross but the hosts took just four minutes to level.
Back in the West Ham side after a decade away, Cole marked his second debut a day after signing from Liverpool by creating the equalizer.
The midfielder whipped in a cross that Collins headed past goalkeeper David De Gea.
West Ham's second was a carbon copy in the 59th, with Collins evading Nemanja Vidic to plant the ball in the net.
But in stoppage time, Van Persie connected with a pass from Ryan Giggs with a deft touch and then beat James Tomkins before striking through goalkeeper Jussi Jaaskelainen's legs.
It was far more comfortable for crosstown rival City — despite the team's preparations being disrupted in midweek by manager Roberto Di Matteo's training ground altercation with striker Mario Balotelli, who started from the bench against Watford.
Instead, Carlos Tevez helped to steer City into the fourth round, netting from a free kick after 25 minutes before a header from Gareth Barry and a late strike from Marcos Lopes on his debut.
In a rematch of the 1987 final, Coventry's hopes of another cup win over Tottenham were dashed very quickly.
Clint Dempsey scored from Gareth Bale's flick-on after 12 minutes and his deflected shot was then pounced on by the Wales winger for the second. A cross from Bale was headed home by Dempsey for the American's second goal.
Three Premier League clubs face replays after being held by lower-league opposition. Wigan drew 1-1 with Bournemouth, Fulham was held by the same score by Blackpool, and Sunderland drew 2-2 with Bolton.
And there will also be a replay between Premier League rivals Queens Park Rangers and West Bromwich Albion after they drew 1-1 in front of just 8,984 fans at Loftus Road.
Aston Villa had to come from behind to oust Ipswich of the League Championship 2-1.
The lowest ranked team left in the cup saw its campaign end as seventh-tier side Hasting went down 4-1 at Middlesbrough.