Notts were boosted at the top of the order by a half century from Alex Hales (55) as the hosts kept wickets in hand.
With only the loss of Hales and Michael Lumb (21), the hosts were able to press on and a partnership of 79 between Riki Wessels (62 not out) and Samit Patel (53 not out) guided the Outlaws to an imposing total of 197-2 from their 20 overs.
The Royals got their reply off to a great start with Vince and Michael Carberry firing the visitors past 50 without loss inside five overs, keeping up with the required run rate in the process.
However, in their attempts to stay on course Carberry danced down the wicket to Patel but was nowhere near the flight of the ball, leaving Chris Read with the easy task of stumping the England opener.
Hampshire were able to regroup as Vince (93 off 51 balls) and Sean Ervine (39) shared a partnership of 83 and Vince was there at the end for the visitors to seal a dramatic victory.