Gloucestershire's players protested in vain after they were given a six-run penalty for a slow over rate.
That left Sussex needing only seven runs off the final over, which Rawlins duly knocked off.
Ian Cockbain's unbeaten 61 had helped Gloucestershire to 159 for six in their 20 overs.
Sussex remain two points behind South Group leaders Kent, who beat Hampshire by four wickets with one ball remaining.
Aaron Milne was the Kent hero with bat and ball, taking three for 21 before hitting the winning runs.
James Vince made 44 as Hampshire were bowled out for 135 in 19.2 overs, but Heino Kuhn put Kent on course for victory as he hit 41 off 33 balls.
Defending champions Worcestershire suffered a surprise defeat by 33 runs to Leicestershire, with the Foxes earning only their second win of the season.
Arron Lilley made 31 off just 12 balls as Leicestershire were bowled out for 152, Wayne Parnell taking four for 25, but Worcestershire never looked like chasing that down as they were always behind on the run rate and wickets fell at regular intervals.
Ben Cox made 30 off 29 balls but was the only player to progress beyond the teens as the Royals were skittled for 119.
AB de Villiers hit a superb unbeaten 88 – taking him past 8,000 career T20 runs – as Middlesex comfortably beat Somerset by 35 runs.
De Villiers hit nine sixes and just one four as he reached 88 off just 35 balls, helping his team to a winning total of 215 for four.
Tom Banton and Tom Abell each made 41 in reply but Somerset's lower order crumbled as they were bowled out for 180 in 17.2 overs.
Yorkshire and Birmingham played out a tie at Headingley as the Vikings' pursuit of 178 fell short at the last.
Tom Kohler-Cadmore made 76 off 54 balls as Yorkshire looked well on track for their target, scoring 88 off the first 9.2 overs before Adam Lyth went for 40.
But left needing two off the final ball, Kohler-Cadmore could only manage a single to mid-off against Fidel Edwards.