Curran, who signed a new contract at Surrey on Friday, claimed two wickets and batsman Vince made an impressive unbeaten 43 as the Sixers chased down their revised target of 84.
The fifth-placed Thunder - needing to beat their rivals at the Sydney Cricket Ground to keep alive realistic hopes of making it through to the semi-finals - were restricted to 128 for six, Chris Green top-scorer on 34.
Following rain delays which saw a revised victory total of 84 from 12 overs, Daniel Hughes and Vince, who last month signed a new four-year deal with county side Hampshire, both produced some big hits to see the Sixers home and move up to second in the table.
After Thunder lost opener Jason Sangha for five, captain Shane Watson, looking well set on a quick-fire 28, edged Curran behind.
Callum Ferguson (36) and then opener Anton Devcich (22) both fell in the space of two overs, with Curran taking another late wicket of Daniel Sams (two).
Curran then produced a brilliant diving take to dismiss fellow Englishman Chris Jordan for one before the Thunder closed out their innings with what looked a beatable total.
Sean Abbott was the pick of the Sixers bowlers with two for 23 from his four overs, while Curran took two for 32.
Rain caused a delay which set an initial revised target of 124 in 19 overs. The Sixers, though, got off to a disastrous start when Josh Philippe was run out on the very first ball of their reply following a mix-up with Hughes.
However, Moises Henriques' men regrouped as Vince weighed in with some big hits before more rain delays in the fourth over forced the players off again, the total at 29 for one.
Play resumed with a revised target of 84 from eight overs remaining. Hughes, finishing 41 not out, also set about the Thunder bowlers again as he pushed on alongside Vince, who hit the winning boundary through point.
The Thunder must now win their last fixture against Hobart Hurricances, while also hoping the Melbourne Stars lose all of their last three matches and Adelaide Strikers suffer at least one more defeat.
