The 33-year-old opener blasted 10 fours and three sixes to lead his side to 341 for seven after winning the toss. David Wiese hit 58 not out and Michael Burgess 56, while there were three wickets each for Max Waller and Tim Groenewald.
In reply, Somerset made 266 all out, the in-form James Hildreth top-scoring with 87. Aussie Matt Renshaw made 55 on his List A debut for the county and Steve Davies 56, but despite their efforts the hosts finished well short.
Danny Briggs turned the match in Sussex’s favour, removing Hildreth and Craig Overton in the 40th over, with the outcome apparently in the balance. Ishant Sharma was the most successful bowler with three for 47.
It was Somerset’s first defeat in any form of cricket this season and they could have no complaints, having been under pressure from the opening overs of the game.
After Luke Wells had fallen cheaply to Craig Overton, Wright collected his first maximum off the same bowler and was soon timing the ball sweetly on a true pitch in bright sunshine.
He found an able partner in Harry Finch and the pair accelerated in good style, putting on 107 in 16 overs before leg-spinner Waller ended the stand by dismissing Finch lbw for 35.
Visiting skipper Ben Brown fell for only eight, but Wright blossomed to take the total past 200 before falling to a well-judged catch by Tom Banton at deep mid-wicket off Waller.
Laurie Evans contributed a bright 43 before also departing leg-before to Waller, whose handy mid-innings spell brought three for 52 from ten overs.
But Sussex needed a big score in the conditions and they got it thanks to Wiese and Burgess, whose half-centuries were full of aggressive, cleanly struck shots.
Still Somerset would have fancied their chances at the halfway stage. But they lost Johann Myburgh and Peter Trego, captaining the side in the absence of the injured Tom Abell, with only 24 on the board, both to Oli Robinson.
Hildreth, a centurion in the previous game against Glamorgan, again looked in fine form. But Steve Davies, having been dropped twice on the way to scoring 56, was unlucky to play a ball from Abi Sakande onto his stumps and when Tom Banton fell to a poor shot the home side were 101 for four.
Hildreth went to fifty off 62 balls and Renshaw hit the ball sweetly to reach a half-century off just 36 deliveries before being caught behind flashing at a wide ball from Wiese.
The departure of Hildreth proved decisive. He was caught and bowled by Briggs off a leading edge and when both Overton brothers, Craig and Jamie, went in quick succession Sussex were as good as home.
Roelof van der Merwe was unbeaten on 34 at the end, but his side were well beaten.
Sussex centurion Wright said: "We knew we would need a big score against a very good side. Fortunately, it was my day today, but the plan was to bat through the innings and as it turned out we needed David Wiese and Michael Burgess to play really well at the end.
"I'm not sure we could have defended 300 on that pitch because Somerset have some quality batsmen, but 340 made a big difference. James Hildreth played superbly and is in great form so we had to bowl well.
"Although we won by 75 runs in the end, it didn't feel like that comfortable a victory. Hildreth's was the big wicket for us."