Having resumed their second innings on 171 for seven at the start of the final day, the hosts slipped to 181 for nine, just 143 ahead, as Mitch Claydon (four for 66) completed match figures of nine for 112.
But 21-year-old Bartlett (63) and Brooks, who marked his Championship debut for Somerset with a breezy 35 not out, took the total to 243 all out, leaving Kent a tricky target of 206 on a pitch still assisting the seam bowlers.
It never looked on, Lewis Gregory claiming three for eight before lunch as the visitors slumped to 41 for five. They were eventually bowled out for 131, Gregory ending with five for 18.
Somerset took 19 points from the game, despite the failure of their top order batting, while Kent gained just four from a match that appeared under their control for long periods.
When Craig Overton and Josh Davey fell early in the morning session, newly-promoted Kent looked to be closing in on a morale-boosting victory over last season’s Championship runners-up.
But Brooks went on the counter-attack, hitting 5 fours and a six in his 31-ball innings, while Bartlett moved to the only half-century of the match, compiled with excellent temperament off 94 balls.
By the time Bartlett was last man out, all the momentum was with Somerset and their bowlers wasted no time capitalising.
Sean Dickson fell to the first ball of Kent’s second innings, edging to third slip off Gregory, who followed up by sending back Daniel Bell-Drummond and Zak Crawley in a superb seven-over spell from the River End.
Brookes weighed in with the key wicket of Matt Renshaw, caught by Gregory at third slip with only a single on the board, while Craig Overton had Heino Kuhn lbw for a duck, with the Kent skipper pushing well forward.
It was 43 for five at lunch and the inspired Gregory struck again in the second over after the interval, Overton taking his second catch at third slip to send back Ollie Robinson.
Alex Blake and Darren Stevens added 37 for the seventh wicket before Overton, bowling around the wicket, had Blake caught behind for 20 with 124 still needed.
Harry Podmore fell lbw, aiming to play Brooks through the leg side, to make it 93 for eight. Matt Milnes fell for ten to a catch at first slip by James Hildreth, giving Gregory a deserved fifth wicket, and in the same over last man Claydon was dropped by Marcus Trescothick at second slip.
It mattered little as Stevens, rock solid throughout his innings of 43, having come in as low as number eight, ran out of partners in the next over, Claydon giving a low catch to Hildreth off Davey.
Somerset’s five-wicket hero Lewis Gregory said: “Momentum is a massive thing in cricket and the partnership between George Bartlett and Jack Brooks at the end of our second innings probably helped us get off to such a good start when Kent batted.
“Taking a wicket with my first ball was a bit strange. It built on what George and Jack had done and we fancied our chances because the wicket was doing a bit all through the game.
“It’s great to start the season with a win and particularly having had to fight so hard for it. We know there will be more tough sessions over the course of a long season and that the character is there in the team to bounce back from them.”
Kent assistant coach Allan Donald said: “There were a lot of positives to take out of the game for us. Without Joe Denly and Sam Billings, we still ran a very good side close and I was delighted with the way we bowled for most of the match.
“Somerset built pressure in our second innings and you don’t come back from 50 for six. They showed what a classy First Division side they are by fighting back and turning the game around.
“Their last wicket stand really hurt us and we lost our way a bit during that period. Had we taken all three remaining wickets early we might have been chasing 140 and instead it was over 200.
“I still thought our target was gettable. But they came out and bowled magnificently.”