Both sides struggled to create any real chance of note during the opening 45 minutes, with neither goalkeeper called into any telling action.
Ba went closest for Chelsea when his low shot was saved by the feet of David de Gea, while for the visitors Javier Hernandez's swerving effort from distance forced Petr Cech into making an unorthodox save with his right boot.
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While there was very few openings during the first half, it took the hosts just three minutes after the restart to break the deadlock. Juan Mata picked out Ba inside the area with a lofted pass and the January signing from Newcastle United beat De Gea with an instant volley into the top left hand corner of the net.
At the other end Hernandez and Cech continued their battle on the hour mark, with the Chelsea keeper coming out on top for a second time. It appeared that the United striker was going to head his side level, but Cech stuck out a hand to send the ball over the bar.
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Three minutes later De Gea matched Cech's reflexes when he got a hand to Mata's powerful shot from a narrow angle.
Having fallen behind, United introduced Robin van Persie into the action and the Dutch international had his side's best chance to level up proceedings four minutes from time.
Patrice Evra picked him out all alone inside the area with a cross from the left, but United's leading goalscorer sent his volley off target.
Chelsea will now face Manchester City for a place in the final later this month.