Resuming on 200 for five, England were dismissed 40 minutes into the afternoon session moments after guaranteeing the tourists would bat again.
Jos Buttler made sure England got past their first target of 298, making his highest score of the series before he was last man out.
Stokes started promisingly, a couple of well-judged leaves off Pat Cummins followed by an authoritative pull for four.
He and Bairstow took 12 off the last six overs of the old ball, after which the real battle commenced.
Starc, largely disappointing on Friday, immediately found the fresh Dukes to his liking and got it swinging right away.
He was able to sow some doubt in Stokes' mind when an inside edge cannoned past leg stump for four and followed up with a full ball that was pushed back at catching height. Starc got his hands to it but could not hang on in his follow through, reprieving England's talisman on 19.
The left-armer was hitting a groove now and soon got his rewards. After sliding a couple across Bairstow he summoned a full inswinger, tailing into the healthy gap between bat and pad to knock over middle and leg.
Bairstow has enjoyed an unhappy series with the bat following an excellent World Cup, an experience his replacement at the crease, Buttler, can readily indentify with.
He got off the mark with a boundary but it was a sign of weakness as much as one of intent, an uppish drive briefly interesting Matthew Wade at short cover.
Australia probably would have been happy to leave Stokes to his business and continue robbing him of partners, but instead Starc landed the big prize.
Stokes had been trying to get the resistance going, top-edging a hook for four off Josh Hazlewood and using his thigh pad to flick the ball to the ropes at fine leg, but it was not to be.
One tentative push at Starc was all it took, a healthy snick carrying to Steve Smith at second slip. Stokes was angry with himself, his latest attempt at digging England out a sizeable hole having faltered.
Jofra Archer was next to go, almost running himself out first ball then caught behind for one wafting wildly at Cummins. England were still 42 runs away from the follow-on mark but Buttler began eating into that, helping himself to three boundaries in six deliveries with Cummins contributing a set of five wides.
England took lunch on 278 for eight, stretching that by five runs before Starc cleaned up Stuart Broad for his third success. Jack Leach would have been lbw seven runs short of the follow-on mark but Australia had frittered their reviews, leaving Cummins to wrap things up with a fast yorker that burrowed through Buttler's drive.