Harlequins looked as if they would respond to last week's embarrassing 52-7 loss to London rivals Saracens with an emphatic win of their own as they surged into a 33-3 half-time lead in front of their own supporters at the Stoop.
But Bath hit back with 33 unanswered points of their own in the last half hour after Harlequins centre Andre Esterhuizen had marked his 30th birthday with a second try of the match.
Bath, in securing a bonus point, leapfrogged champions Saracens into second place in the table, with Quins climbing into fourth spot. Only the top four teams at the end of the regular season qualify for the title-deciding play-offs.
"First and foremost, we had a pretty big week together, bouncing back from that (defeat at Saracens) last week, so full credit and I'm so proud of the boys and the whole staff to get it right," said Quins boss Billy Millard.
"We bounced back and I thought in the first half we played some of our best footy."
He added: "The second half, I think there was a bit of ill-discipline, yellow cards (for Irne Herbst and Louis Lynagh), a momentum shift and they got on the front foot. But I'm just really proud of the boys to bounce back like that after a tricky week."
Harlequins dominated the first half with Esterhuizen scoring his first try inside three minutes when sent in by scrum-half Danny Care, playing his first game since announcing his retirement from England duty.
Scotland star Finn Russell landed a penalty to put Bath on the scoreboard but from then it was one-way traffic for the remainder of the opening period.
Smith, in a moment of typical individual skill, chipped over the Bath defence and regathered for a fine try before Italy wing Lynagh, the son of Australia great Michael Lynagh, went over in the corner.
England back-row Alex Dombrandt's catch-and-drive try ensured Harlequins had a bonus point within 30 minutes and flanker Will Evans added another try before half-time.
When Smith converted Esterhuizen's second try, Quins led 40-3 only for Bath to launch an extraordinary fightback.
Tries from Will Muir, Alfie Barbeary, Louis Schreuder, Ruaridh McConnochie and Elliott Stooke, with Russell converting four of those scores, set up a grandstand finish as Quins had two players sent to the sin-bin.
Harlequins, however, held out in the closing minutes after Stooke's converted try cut what had been a commanding lead to a mere four points.
Elsewhere, Bristol maintained their play-off bid by coming from behind to win 33-24 away to southwest rivals Gloucester.
Magnus Bradbury's two tries plus scores for Fitz Harding, Gabriel Oghre and Benhard Janse van Rensburg were the standout moments for Bristol as they came from 24-19 down at Kingsholm.
