Care walked out ahead of the rest of his team-mates with his children for his final appearance at The Stoop, as all sides of the famous ground gave him an emotional standing ovation.
His team-mates needed just 175 seconds to open the scoring as Cadan Murley went over in the left corner thanks to Rodrigo Isgro’s long pass.
Exeter had Tamati Tua sin-binned four a 14th-minute head-on-head clash with Quins skipper Alex Dombrandt as the hosts had 85% of the possession in the first quarter.
They eventually made it tell when starting scrum-half Will Porter – wearing the 21 shirt – darted around the short side of a 21st-minute line-out maul.
But just after the kick-off Exeter responded as full-back Josh Hodge – one of the few bright lights in their disappointing season – broke through the Quins defence and Rus Tuima was on hand to power over from close range despite being a man down.
Seven minutes later the returning Tommy Wyatt got on the end of a line-out maul for his fourth Exeter try of the season, but Henry Slade missed the conversion for the Chiefs, who had new backs coach Dave Walder in attendance for the first time.
Exeter continued to grow and they went ahead three minutes into the second half when Stu Townsend capitalised on a Wyatt break as the struggling Devon side showed some much-needed confidence.
The visitors did well to repel a number of dangerous Quins attacks as the half wore on – a vast improvement in the past few weeks which culminated in a disastrous 79-17 loss last month at Gloucester that cost coaches Rob Hunter and Ali Hepher their jobs.
But Exeter could do nothing to stop the individual brilliance of England’s Smith as the Quins fly-half celebrated his British and Irish Lions call-up with a phenomenal individual try.
With 20 minutes to go he skipped past five tacklers and raced in under the posts as his conversation made it 21-19.
A Slade penalty soon after put Exeter a point ahead before Care came onto the field for one last time to the joy of the home fans.
They saw their side go back in front when Smith kicked a penalty from in front of the posts.
Quins came close to scoring a fourth try when Murley broke through with two minutes left and fed former Chief Tom Lawday, but from the resulting breakdown Wyatt picked up the ball in an offside position.
For a moment it seemed as though Care would try and a kick a penalty to end his career.
However, with too much time left on the clock Quins opted against it and settled on keeping possession to seal the win which moved them up to seventh place.
