Ibrahim Zadran (32 off 32 balls) and Usman Ghani (30 off 20 balls) anchored the innings, but Curran ran through the middle and lower order as Afghanistan lost 5-3 in 12 deliveries at the death in Saturday's clash.
Curran's haul was the first time an England player has snared five wickets in a T20 international.
"At the start of the day I did not expect to be walking off like that," Curran told Fox Sports after the haul.
"We are really happy with the score."
The 24-year-old's heroics came after England's fielders pulled off a series of stunning catches earlier in the innings.
Liam Livingstone got the small crowd going with a fantastic diving catch running in from deep cover to remove opener Hazratullah Zazai (seven off 17 balls).
Ibrahmi Zadran was lucky to survive on 16 when England fluffed a run-out chance, and he added a further 16 runs before Moeen Ali kept his composure to catch a tricky skied ball.
Adil Rashid went one better with his fingertip effort running with the flight to dismiss Najibullah Zadran (13).
But nothing compared to the effort of wicketkeeper captain Jos Buttler, who dived full length to take a superb one-handed catch down the leg side to dismiss Mohammad Nabi.
Chris Woakes took a reflex one-handed catch off his own bowling for good measure, but it was the Curran show late as the left-arm pace star ended any hopes of Afghanistan adding some late lusty blows.
Curran was full of praise for the team's fielding efforts, with all 10 of Afghanistan's wickets falling to catches.
"We've worked hard on our fielding over the last couple of weeks," he said.
"We know it's going to be a huge factor in the tournament.
"Livingstone started us off with that running catch in front of the Barmy Army. That set the standard for the rest of the innings."
England suffered a huge blow earlier this week when pace bowler Reece Topley was ruled out of the tournament after injuring his ankle while stepping on a boundary advertisement cushion during a catching drill.
It came after fellow England star Jonny Bairstow was ruled out last month after breaking his leg in a freak golfing accident.