The Blues, who have lifted the trophy in four of the last six seasons, will now face Manchester City at Wembley in the semi-finals - with another fixture added to an already hectic April schedule for Rafa Benitez's side.
Fatigue had been a buzz word prior to kick-off, with the games coming thick and fast at this stage of the season, and a lethargic opening from both sets of players suggested Robin van Persie's pre-match warning not to expect a classic would be proved correct - with misplaced passes and heavy touches the order of the day.
It took over half an hour for the first meaningful effort on goal to be mustered, with Ba forcing David de Gea into a low stop with his feet.
Prior to that, the sight of Ashley Cole failing to keep pace with England colleague Danny Welbeck and subsequently limping off with a hamstring complaint had provided the only talking point for those who had braved the capital cold on Easter Monday.
Things did improve marginally as the afternoon wore on, but David Luiz fired a 25-yard free-kick high into the stands and Nani saw a dipping effort from distance skip off the turf and wide of the target.
Javier Hernandez, a man who has tormented Chelsea on a regular basis in the past, came as close as anyone to forcing a first-half breakthrough shortly before the interval, with Petr Cech sticking out a boot to fend off an ambitious strike which moved wildly in flight.
In a game lacking any real quality, it appeared as though a moment of magic, or possibly madness, would be required to settle the tie, and Ba provided the former on 49 minutes.
Juan Mata teased a hanging ball into the box and, after watching it drop over his shoulder, Chelsea's Senegalese striker swivelled on the slide to send a looping volley over De Gea and into the back of the net.
With the likes of Van Persie, Ryan Giggs, Frank Lampard and Fernando Torres waiting patiently on the bench, and with recent contests between Chelsea and United having produced a flood of goals, you got the feeling that one was unlikely to be enough for the hosts.
Chelsea were, however, by hook or by crook, able to contain United - with Cech producing a remarkable fingertip stop on the hour mark to divert a diving header from Hernandez up and over the crossbar.
Shortly after, Mata was presented with an opportunity to try and double the Blues' advantage, but he took too long to set himself and eventually saw a fierce drive from an ever-tightening angle touched into the side-netting by De Gea.
Eden Hazard then took it upon himself to try and ease the tension around Stamford Bridge but, after bursting through the middle and into the penalty area, he could only slide wide of the post when offered a clear sight of goal.
United were expected to try and finish with a flourish, in an effort to keep their double dreams alive, but they created only one opportunity of note - which Van Persie hammered high into the crowd - and Sir Alex Ferguson's men must look to console themselves with a bid to take the Premier League title with a new record points haul.