The home side, chasing 378 to win, were bowled out for 285 just before lunch.
Having resumed on 181-4, their hopes were undermined by the early dismissals of Will Smith and Adam Wheater.
Off-spinner Luke Wells (2-24) claimed the vital wicket of Sean Ervine for 42 and bowled last man James Tomlinson to bring the game to an end.
The three-times champions, third behind Yorkshire last summer, needed less than 20 minutes to remove Hampshire's overnight pair
And it was Eastbourne-born seamer Matthew Hobden who did the early damage as he had Smith caught behind down the leg-side and bowled Wheater in successive overs.
Paceman Ajmal Shahzad (3-63) completed a promising Sussex debut by removing Gareth Berg, who prodded to short extra cover, before Wells had Ervine smartly taken by Craig Cachopa at leg-slip to reduce the hosts to 244-8.
Chris Wood hit a straight six off Wells and two fours in his innings of 30, but then edged a ball from Steve Magoffin to the wicketkeeper and Wells completed the victory three overs later.
Hampshire skipper Jimmy Adams:
"I thought yesterday guys showed a lot of character and gave us a chance of winning the game today - but it wasn't to be.
"We missed a big score from one of the top six, which was the difference in the two teams."
Sussex captain Ed Joyce:
"It is great to get a win under our belts against a really good team.
"We let ourselves down with the bat in the second innings to give them a sniff of victory, but our bowlers throughout the game were outstanding and so was our catching - both of those things helped us win the game.
"Hampshire were a really strong team, even without their overseas bowler (Jackson Bird). They will be a difficult side to beat this season."
Sussex underlined the gap between the divisions in the County Championship by completing a 92-run win over promoted Hampshire at the Ageas Bowl.