Curran has developed into a reliable option at the death and his four wickets in six balls to finish with figures of 5-10 polished off Afghanistan for just 112 in 19.4 overs.
England made hard work of the total, stifled by Afghanistan's vaunted spinners, but the returning Liam Livingstone's unbeaten 29 off 21 balls helped to secure the win with 11 deliveries to spare.
Livingstone, taking part in his first competitive match since late August after an ankle injury, also took an outstanding catch, as did Moeen Ali, Adil Rashid and Jos Buttler in a tremendous collective fielding effort from England.
Afghanistan were on 82-3 after 14 overs but collapsed dramatically, as Ben Stokes finished with 2-19 while fast Mark Wood also collected two wickets and averaged 92.6mph on a springy pitch with plenty of carry - the fastest four-over bowling performance in this tournament's history.
The absence of Reece Topley to a tournament-ending ankle injury this week did not impact England here and they may have been grateful for going batter-heavy in selecting both Livingstone and Harry Brook in a stuttering run-chase, with Chris Woakes passed fit despite concerns about a tight quad this week.
After England won the toss, Stokes was entrusted to open the bowling in his first outing at this tournament since the 2016 final before Wood made the breakthrough with his first delivery as Rahmanullah Gurbaz (10) was caught on the crease and tickled through to Buttler.
Wood touched a breakneck 96mph with his next delivery and while Ibrahim Zadran showed a rare show of force by hooking the Durham man for six, he then fenced at another four deliveries.
Afghanistan were going nowhere fast and their cause was not helped as Livingstone took a blinder, running in from deep point and swooping to pouch a low chance to see off Hazratullah Zazai, who had crawled to seven off 17 balls before hacking at Stokes.
Curran had his first as an off-cutter was miscued by Ibrahim, who top-scored for Afghanistan with a run-a-ball 32 after Moeen took a catch over his shoulder, while Najibullah Zadran (13) departed after Rashid shuffled back from midwicket and took a low chance lunging forward to give Stokes another wicket.
England captain Buttler took a fantastic leaping catch down the leg-side to see off opposite number Mohammad Nabi (3), who gloved behind off Wood, while Afghanistan belatedly reached three figures in the 17th over.
Curran had two in two as Azmatullah Omarzai and Rashid Khan were caught in the deep and while the left-arm seamer missed out on a hat-trick, he had three wickets in four balls when Usman Ghani slashed to deep backward point.
Curran then wrapped up the innings when last man Fazalhaq Farooqi picked out Dawid Malan.
Buttler took authoritative fours in each of the first three overs of England's chase but perished for a run-a-ball 18 after flicking Farooqi to deep midwicket, moments after Hales was dropped on four.
England were in the driving seat on 40-1 after the powerplay but the brakes were put on by spinners Rashid Khan and Mujeeb Ur Rahman, who had Hales spilled again off his bowling on 17.
Hales' luck ran out as he pulled to deep backward square-leg for 19, Stokes was beaten for pace as he charged at off-spinner Nabi and was bowled through the gate, while the Afghanistan captain dived to snaffle Malan, out for 18 off 30 balls in a becalmed innings which did not feature a single boundary.
Livingstone was busy at the crease and took back-to-back offside fours off Fareed Ahmad but Brook slammed to long-on in Rashid's final over as the leg-spinner finished with one for 17.
Livingstone and Moeen took no risks from that point forwards, rotating the strike to get England over the line in the penultimate over. They next take on Ireland in Melbourne on Wednesday.