Eric Elwood's men ruined Quins' hopes of reaching the knockout stages last January with a nail-biting 9-8 defeat at the Galway venue.
Nine months on, both sides threw off the shackles to share out 43 points in a free-flowing first half full of enterprising rugby.
Dan Parks converted David McSharry's seventh-minute try and added four penalties to move the hosts 19-9 ahead.
But Quins grabbed 15 points in the closing 11 minutes before the break, that profitable spell being crowned by Care's two shrewdly-taken tries.
Ben Botica, an excellent replacement for Nick Evans against Biarritz in the opening round, landed an injury-time penalty to complete a 24-19 turnaround by the interval.
Botica turned in another dazzling place-kicking display with a 20-point haul, as two further three-pointers from him killed off Connacht's doughty challenge.
The Irish side had settled quickly with a fine touchfinder from Parks and a lineout steal from Mike McCarthy, but the visitors were the ones to show the early adventure.
Sam Smith and Mike Brown danced into the Connacht half, Care's backhanded pass released Maurie Fa'asavalu and the result was a first penalty success from Botica.
Connacht hit back within two minutes, George Naoupu taking a quick tap and Parks' flat pass sent McSharry motoring through a gap between Jordan Turner-Hall and Matt Hopper and in behind the posts.
Parks' simple conversion was cancelled out by Botica's second three-pointer, following some more quick-witted attacking from Quins.
The English champions were looking the more potent side, but the wind-assisted Parks punched off a terrific 52-metre penalty to make it 10-6.
This feisty encounter was turning into a fascinating duel between the out-halves as tournament newcomer Botica answered back in the 16th minute.
Connacht continued to make a flurry of tackles, their industry up front setting up a 45-metre penalty that Parks brilliantly bent through the posts.
Quins blundered as Turner-Hall got isolated in a promising position and then after a smashing break from Kieran Marmion, the visitors infringed at a ruck in front of their posts as Parks stretched the margin to seven points.
The former Scotland international then nudged over his third penalty on the trot, set up by Fetu'u Vainikolo's midfield snipe and further input from John Muldoon and McCarthy.
But the hosts were rocked by the concession of a try on the half hour mark as man-of-the-match Care broke away from a scrum and evaded Muldoon's grasp on the way to the line.
The unconverted effort spurred Conor O'Shea's side on and Care nabbed his second seven minutes later, dummying to his right from a close-in ruck and wriggling in past Muldoon again.
Botica converted this time and his penalty in added time left Connacht five adrift at the break, with the tide turning sharply against them.
McCarthy muscled forward to create the opening for Parks' fifth penalty of the night as the tempo dropped briefly in the third quarter.
Botica maintained his high standards from the kicking tee with a sure-footed 50-metre strike, and he split the posts from closer in after Adrian Flavin infringed.
Chris Robshaw and Nick Easter were to the fore as Quins tightened their grip on the game in the closing stages, their second pool win pushing them four points clear of Biarritz at the top of Pool 3.
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