Rain led to a delayed start at the Oval, but the match, which was reduced to 10 overs per side, eventually got under way in light drizzle at 8.20pm.
Having won the toss, Surrey posted 84 for four. Kevin Pietersen, making first appearance in the competition since announcing his retirement from all one-day international cricket, was brilliantly caught at extra cover off the first ball.
But Steven Davies gave the hosts a platform from which to build by clubbing 30 in 11 deliveries, including six fours, before picking out Chris Wood at mid-off in the third over.
Surrey were pegged back further in the fifth over, when Jason Roy was caught at wide long-on to hand Liam Dawson a second wicket.
With Zander de Bruyn only managing to score eight off 14 balls before being bowled around his legs by Wood, the hosts were indebted to Zafar Ansari's unbeaten 35 in 25 deliveries, which included five boundaries.
In reply, the Hampshire Royals made 63 without loss in 5.4 overs.
By the end of the fifth over James Vince, in tandem with Jimmy Adams, had taken the game by the scruff of the neck, leaving Hampshire needing 34 off five overs. Adams lifted Stuart Meaker over mid-wicket for six, but with the rain intensifying the contest was halted shortly after the ball slipped out of Meaker's hand and flew to the boundary for four no-balls.
With no prospect of getting the game restarted, Adams and Vince finished undefeated with 28 and 24 respectively, having both faced 18 balls. The result means that Surrey, who have now suffered four successive defeats in the FLt20, remain rooted to the bottom of the South Group table.
Meanwhile, Gloucestershire and Glamorgan suffered huge frustration as their Friends Life t20 match at Bristol was abandoned without a ball bowled following a 6.30pm pitch inspection.