The Blues looked in danger of losing their grip on the trophy when Jay Rodriguez fired Saints ahead in the 22nd minute but Ba and Victor Moses replied to put Chelsea 2-1 up at half-time.
Branislav Ivanovic, Ba, who started up front in place of Fernando Torres following his £7m move from Newcastle on Friday, and Frank Lampard then struck after the break as the visitors cruised to victory against their Premier League rivals.
Southampton made a bright start and it required a timely intervention from Ivanovic to check Rodriguez's early charge forward.
David Luiz sliced over from 25 yards at the other end before Saints goalkeeper Artur Boruc raced smartly off his line to deny Ba.
Both sides were passing the ball around well but it was the hosts who went ahead midway through the first half when Jason Puncheon broke into midfield and slipped in Rodriguez, who poked the ball past Ross Turnbull.
Chelsea hit back and Eden Hazard fizzed a shot just wide of the far post before Ba levelled in the 35th minute.
Moses' delightful flick sent Hazard clear down the left and he then fed Juan Mata whose clipped effort was kept out by Boruc's fingertips, allowing Ba to slam home the loose ball.
Moses then saw a fierce effort saved by Boruc before putting the visitors ahead seconds before the break by rifling home a low shot after good work by Ashley Cole down the left.
Chelsea remained in charge after the restart and extended their lead in the 52nd minute when Mata picked out Ivanovic with a superb cross from the left and the defender powered home a close-range header for his sixth goal of the season.
Senegal international Ba then made it 4-1 in the 61st minute with a first-time shot at the near-post to convert Hazard's slide-rule pass.
Southampton's misery deepened when defender Jose Fonte was stretchered off and they were also denied a goal as Turnbull made a superb double save to foil Steve De Ridder and Steven Davis.
Ba then went close to completing a hat-trick but his header from Moses' cross was well saved by Boruc with his feet.
Chelsea were awarded a penalty with seven minutes remaining after Davis had handled Lampard's lifted pass and the veteran midfielder - rather than Ba - stepped up to convert the spot-kick to equal Bobby Tambling's record of 25 FA Cup goals for the club.
The goal also made Lampard Chelsea's joint second highest goalscorer of all time alongside Kerry Dixon on 193 goals - Tambling leads the way with 202.