The quarterfinal replay came to life four minutes into the second half when Ba exquisitely hooked the ball over goalkeeper David De Gea to set up a Wembley Stadium meeting with Manchester City.
"It was a brilliant pass from Juan," Ba said. "For Newcastle, I scored quite a similar goal against Man United. It reminded me of that."
United was denied an equalizer at Stamford Bridge when Petr Cech pulled off a stunning one-handed save to block Javier Hernandez's close-range header.
While Chelsea also has a Europa League quarterfinal on Thursday, United is left only with the Premier League in which it holds a seemingly insurmountable 15-point lead over City. The two Manchester teams play Monday.
"It was disappointing today, of course, but we have to forget all about it," United manager Ferguson said. "We are in a good position obviously, and I think a positive result for us on Monday will just about seal the title for us."
Advancing in the cup was the perfect response by Chelsea to Saturday's loss at Southampton, which dropped the team to fourth in the league.
Although Chelsea is on course for a fifth FA Cup triumph in seven seasons, the priority for the club is still securing a top-four finish and Champions League qualification. However, the win puts interim manager Rafa Benitez, who is only under contract until the end of the season, closer to getting a trophy to show for his short spell at Stamford Bridge.
"The priority is the top four ... the race for the top four will be until the end of the season," Benitez said. "We're in the top four with a game in hand. We're in a very good position, the semifinals of the FA Cup and the quarterfinals of the Europa League."
With the fixtures piling up, Benitez has to manage his squad, with Ashley Cole joining fellow defender Gary Cahill on the sidelines Monday.
Cole, who was rested on Saturday, pulled up after just 20 minutes with a hamstring injury that will keep him out for two weeks. Cahill is set be sidelined for at least 15 more days after injuring his knee before the two-week international break last month.
"We can't kill players playing them two games in a row, because then they get injured and we lose them," Benitez said. "Our squad is not too big but we can manage if we don't have injuries."
Few matches will witness an atmosphere as toxic as this one.
Boos greeted United defender Rio Ferdinand's name when it was read out before kickoff, as was most touches during the game.
The response by United fans was to chant about Chelsea captain John Terry, making clear references to his ban for racially abusing Ferdinand's brother, Anton, during a match at Queens Park Rangers last season.
Terry was on the bench in one of seven changes from the side that lost at Southampton, but neither defence was troubled initially.
There wasn't even a shot on goal for 31 minutes, with Eden Hazard eventually breaking forward and releasing Ba, whose shot was turned away by De Gea's left outstretched foot.
Hazard had a chance after playing a one-two with Oscar but the Belgium winger blasted it over, while Cech scooped a shot from Hernandez clear with his right foot.
There was little De Gea could do to keep out the eventual match-winner.
Mata lobbed a pass over the United defence and Ba raced past Ferdinand and hooked a volley over De Gea while almost sideways in the air.
United pushed for the equalizer but was thwarted by Cech's superb save that even had Hernandez congratulating the Czech Republic goalkeeper.
Antonio Valencia surged forward and whipped in a cross that Hernandez met with a diving header that appeared to be heading into the net before Cech tipped over with his left hand.
United was without striker Wayne Rooney due to a groin injury while Robin van Persie didn't come off the bench until the second half.
Van Persie squandered his best chance to equalize when he made a poor connection with Patrice Evra's late cross, shooting high when unmarked in a central position.
"We lacked the composure to win the game," Ferguson said. "We had a lot of possession but didn't make the best use of it."
Chelsea will now play Manchester City on April 14 at Wembley, a day after Wigan takes on Millwall in the other semifinal.
In Monday's only Premier League game, Queens Park Rangers lost 3-2 at Fulham to be seven points from safety and only off the bottom of the standings due to its superior goal difference over Reading.
Dimitar Berbatov put Fulham ahead from the penalty spot after Christopher Samba had brought down Ashkan Dejagah. A blunder by Samba allowed Berbatov to extend Fulham's lead in the 22nd.
Clint Hill gifted Fulham a third when he turned John Arne Riise's cross into his own goal in the 41st.
QPR fought back with Adel Taarabt pulling one back right on halftime and Loic Remy scoring after the break, but Fulham held on despite Steve Sidwell's dismissal in the 79th.