After being asked to bat first, Sibley and Jacks reached 59-0 at the end of the six-over powerplay, but the stand was broken on 74 in the eighth over when Jacks cut Benny Howell to Liam Dawson for 41 from 24 balls.
In-form Jason Roy was run out following some fine work from the sprawling James Vince after scoring seven from 13 balls, having amassed 258 in his previous five knocks.
With his side 96-2 in the 12th over, Sibley got back to work and brought up his half-century from 32 balls but after adding 42 from 27 balls with Curran, he eventually fell for 70 off 46 balls to a diving catch from James Fuller at mid-on after toe-ending one off Scott Currie (2-36).
Curran and Laurie Evans added 34 from the next 13 deliveries before Curran skied Fuller to Vince to depart for a 20-ball 38 with two overs remaining.
Evans was caught by Dewald Brevis off Currie in the final over for 19 off 10 balls, and the visitors closed on 193-6 with Tom Curran run out off the final ball for 10.
Lhuan-dre Pretorius clipped Sam Curran off his pads for a six from the third ball of Hampshire’s reply and showcased his power with 14 off Reece Topley’s first over on his return to the Surrey side.
However, the South African fell in tame fashion in the fourth over, trying a reverse sweep only to top-edge to Topley (2-24) off Jacks for 28 from 19 balls.
Hawks skipper Vince (9) followed him in the following over, looping Topley to Chris Jordan at mid-on, and Toby Albert inside-edged Nathan Smith behind to Evans as the hosts were 45-3 at the end of the powerplay.
Brevis welcomed Kiwi spinner Mitch Santner with two sixes in three balls, but the game looked up when he holed out to Smith in the deep off Jordan to depart for 16 from 10 balls with 120 runs still needed from 11 overs.
Jordan (2-17) picked up his second wicket when Joe Weatherley’s promising knock ended as he pulled one to Ollie Sykes on 27, while Fuller (11) looped Santner to Jacks, and Liam Dawson (1) chipped Topley to Curran to leave the Hawks 98-7 in the 14th over.
Santner bowled Chris Wood (3) to end with 2-36, and Smith had Benny Howell (10) caught by Roy and John Turner (4) hole out to Jordan to finish with figures of 3-18 as the Hawks were bundled out with 13 balls to spare.