The hosts, who have now dropped to second in the South Group, reached 157-6 in 20 overs after being put into bat by the Sharks, but they struggled to see out a win as Craig Copacha and Matt Machan hit unbeaten knocks of 89 and 56 respectively.
Michael Carberry and captain James Vince opened the batting for the home side after Sussex won the toss and elected to bowl first, and it was the former who grabbed the first boundary of the match by lofting his shot over Oliver Robinson's head.
Vince, who has been named in the England squad for next week's T20 series against New Zealand, held his breath when he tried to pull a shot away as Chris Liddle's effort hit the top edge, but the batsman survived when the ball fell through the hands of the Sussex fielder and trickled to the boundary.
A few balls later, Carberry was given a huge let-off when his lofted shot towards mid-on looked to result in a certain catch for Tymal Mills, but the Sussex man fluffed his opportunity and dropped.
The opening partnership produced 33 runs before Carberry was dismissed for 10 off the bowling of Mills when he was caught at short fine leg after five overs.
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Sussex took control of the field by snatching a third wicket inside 10 overs as Adams (21) was caught behind by Brown off the delivery of Matt Machan, who was the Sharks' sixth bowler of the innings.
Adam Wheater impressed by hitting back-to-back boundaries to push Hampshire to 104-4 after 14 overs, but he could have been sent packing a few balls later as he swung at Mills's delivery and the ball came close to falling into Brown's hands, but the wicketkeeper failed to grasp it.
With three overs remaining, big-hitter Shah survived a brief scare after whipping the ball away to deep mid-wicket and it was dropped short, but the 36-year-old was sent packing shortly afterwards on 36 runs after being caught off a slower ball from Robinson.
At the end of the innings, Wheater contributed with 26 runs, while Will Smith and Gareth Berg, who was playing in his first T20 game for Hampshire after replacing Jackson Bird, finished unbeaten on nine and five runs respectively.
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In reply, Sussex suffered an early blow when Luke Wright (3) cut the ball off Chris Wood and Smith jumped to his left to grab it with an impressive two-handed catch, which immediately swung the momentum in Hampshire's direction.
Wood continued his dominance when he dismissed Mahela Jayawardene for four just five balls after taking out Wright, and the hosts went from strength to strength when Berg's delivery forced Brown into the air, which resulted in a simple catch for Ervine.
At 14-3, Cachopa and Machan were given the task of recovering Sussex's innings, and the former was able to calm the storm by hitting two quick fours, while his partner sat on eight runs off nine balls after the sixth over.
The batting pair steadied the innings, but towards the end of the eighth over Vince was just inches away from a run-out as Cachopa was caught napping, but the Sussex man made up a yard or two after the ball glided past the stumps.
With the Sharks needing to increase the run-rate, Cachopa and Machan were able to hit 19 runs off Danny Briggs as the visitors moved to 73-3 after 11 overs, leaving the hosts slightly rattled.
The contest was perfectly poised after 13 overs as both teams at that stage were at 92/93 runs, but Cachopa showed no signs of slowing down for the visitors as he swept four off Wood to bring up his 50.
Hampshire's strong start looked less meaningful as the Sussex duo brought up a 100-run partnership after 15 overs and to add insult to injury, Cachopa launched the ball out of the stadium for six off Yasir Arafat's opening bowl of the 17th over.
Sussex needed just seven runs off 12 deliveries, which Cachopa and Machan were able to see out comfortably.