The duo both hit their second County Championship tons of the season as they put on 161 runs for the third wicket.
Vince took the aggressive role, hitting 19 boundaries to reach his 100 from just 121 balls before Tom Curran (2-71) had him caught behind for 104.
George Bailey (61 not out) and Adams put on 129 before Adams was bowled in the last over, leaving Hampshire 361-4.
It was a near-perfect innings from left-hander Adams, who hit 19 fours and two sixes, offering no chances until he was dismissed by a full ball - the 262nd he faced - from off-spinner Amar Virdi.
Hampshire won the toss but lost Lewis McManus, opening because of Liam Dawson's England Test call-up, for 13 when he was bowled by an inswinging yorker from Sam Curran.
Rilee Rossouw, on a terrible run of form, made 28 but edged Tom Curran to second slip, before Vince took centre stage, scoring fluently through the offside, with 76 of his 104 runs coming from fours.
Bailey then played a well-judged support role for Adams, getting Hampshire past the new ball before Adams' dismissal with five balls left brought stumps.

Griffiths won a trio of player of the year awards after scoring 40 goals last season but has started only 10 league games this campaign.
He missed Sunday's win over Hearts with a calf injury and Rodgers made it clear the striker must improve his fitness.
"What I've said to him is about making sure the talent is a working talent," said the Celtic manager.
"Because talent's not good enough on its own.
"You have to work at it - and it's a cycle. It's one where, when he's playing and fit, Leigh's very good but can still work harder in training.
"The problem is if you're not getting the games and then if you're not quite on it in training then you're losing fitness.
"But then the moment you're asked to come in and get up to speed in a real high-tempo, intense game, the moments that are there for you to risk injury are far greater."
Rodgers named Dedryck Boyata and Callum McGregor as examples of players who have shown great professionalism to work hard and force their way into his plans.
"You have to work tirelessly every day or unfortunately you'll get left behind," he said.
Rodgers echoed the words of his predecessor, Ronny Deila, who spoke of the need to be a "24-hour athlete".
"It's a difficult period for him with that injury," the Celtic manager said of Griffiths, who has lost his place to summer signing Moussa Dembele.
"But the only way you can overcome that is by working 24 hours a day for your profession, for your life.
"He's a good kid, a great boy, but he understands where he's at. We all know the talent, but it won't be good enough on its own.
"We speak a lot and it's not easy because he hasn't played maybe as much as he did last year. The reasons are simple for that.
"He's a player I really admire, a player we really want to have here.
"He's part of the squad, but I want him fit and available. He knows that and it's something he's working harder now to try to achieve."
Jimmy Adams and James Vince struck centuries to put Hampshire in control on day one against Surrey at the Oval.
[NEXT_CONCEPT]
Brendan Rodgers has urged Leigh Griffiths to work harder if he is to force his way back into Celtic's team.