Cutting's unbeaten 42 off just 17 balls carried Quetta to 190-5 in 19.4 overs to put the title defender on top of the points table with three victories from four games.
Needing 11 runs off the last over, Cutting lofted fast bowler Amad Butt for two straight sixes to seal the victory with two balls to spare as Islamabad surprisingly didn't use young fast bowler Mohammad Musa for the last over.
Cutting had a lucky escape in the penultimate over when he was caught at deep square leg off a no-ball as left-arm seamer Rumman Raees had overstepped.
It was Islamabad's second defeat in the tournament after Quetta beat the two-time champion in the opening match in Karachi.
Quetta fast bowlers Naseem Shah (2-13) and Tymal Mills (2-34) earlier held Islamabad's total to 187-7 with Colin Ingram scoring an unbeaten 63 off 41 balls.
Naseem, who became the youngest fast bowler to record a hat trick in a test match at the same venue against Bangladesh earlier this month, claimed both wickets in his first two overs when he clean bowled Luke Ronchi (11) and Dawid Malan (10).
Captain Shadab Khan (39) was twice let off the hook against fast bowler Mohammad Hasnain when Shane Watson missed a straightforward catch at midwicket and Azam Khan then dropped a sitter at point.
The fielding lapses allowed Shadab to add 76 runs with Ingram for the fourth-wicket stand before Naseem and Mills limited Islamabad in the final overs.
Ingram was also twice dropped before he completed his half century as Quetta conceded just 33 runs in the last five overs.
Watson's struggle with the bat at the top order continued before the Australian holed out at deep mid-wicket for another disappointing knock of 9.
Jason Roy scored a 38-ball 50 and Sarfaraz made 33 but it was Cutting's masterful knock in the end which carried Quetta home. Left-arm spinner Ahmed Safi Abdullah took 2-17 and Faheem Ashraf, who dismissed Watson early, was expensive with 2-46.