The bowling all-rounder became the first Englishman to take five wickets in a T20 international, posting figures of 5-10 as Afghanistan were bowled out for just 112 in Perth.
Instead of attempting to whittle off their target at a pace, England took their time, eventually reaching 113-5 with 11 balls to spare.
After watching group rivals Australia succumb to a heavy defeat to New Zealand, England won the toss in their own game and successfully restricted their opponents.
Afghanistan came through the first half of their innings having lost just two wickets, but only Ibrahim Zadran (32) and Usman Ghani (30) offered much with the bat.
As the innings progressed, Ben Stokes and Mark Wood took two wickets apiece, although it was Curran who stole the headlines during the closing overs.
The 24-year-old ended with remarkable figures of 5-10 off 3.4 overs, an effort which included 16 dot balls, and it set up England to chase down 113.
While each of Jos Buttler and Alex Hales struggled to find consistency at the top of the order, scores of 18 and 19 respectively kept England on track until Stokes was dismissed for just two off four balls, leaving his side on 65-3 off 10.4 overs.