The holders will play Manchester City in the semi-finals after Senegal international Ba, who had only scored once for Chelsea in his previous 10 appearances, struck with astonishing athleticism after 49 minutes.
Ba, drifting free of his marker, allowed a lofted ball from Juan Mata to sail over his shoulder before stretching to volley it into the far corner beyond a helpless David De Gea.
United's Mexican forward Javier Hernandez seemed bound to equalise with a diving header but Cech performed a backward twisting dive to palm the ball over the bar and win the tie for Chelsea who were 2-0 down after 11 minutes in the first match at Old Trafford before fighting back to draw 2-2.
"I scored one that was quite the same for Newcastle against Man United, so when I scored this one it reminded me of it," Ba told ITV. "But it was important to win today, it gives us a lot of confidence."
"We lacked the composure to win the game," he said. "We had a lot of possession but did not make the best use of it and then on the counter-attack they caused us quite a few problems."
Chelsea's semi against City will be their sixth in the last eight seasons and Monday's victory was their 23rd in 28 FA Cup matches.
Chelsea's interim manager Rafa Benitez, who is deeply unpopular with the club's fans and is leaving at the end of the season, said he was looking forward to leading a team out at Wembley for the first time.
"It will be in good, when I was at Liverpool we played the final in Cardiff - and I hope I can get to the final at Wembley too."
