The wet weather left England needing to chase a revised target of 101 in 25 overs, which they did so without too much alarm thanks to an Ian Bell half-century.
An economical Chris Jordan earlier claimed two for 13 as Afghanistan crawled to 111 for seven from 36.2 overs before a third rain delay forced the players off for two and a half hours.
At that stage it appeared the match was destined to be rained off and the points shared, which would have condemned England to their worst-ever World Cup in terms of wins.
They were spared the ignominy of that record as play resumed just 22 minutes before the cut-off time and then openers Bell and Alex Hales put on 83 to make short work of the pursuit.
Hales was twice dropped in almost identical fashion by Najibullah Zadran at backward point - the first from just the third ball of the innings - much to the bemusement of left-arm quick Shapoor Zadran.
Hales, batting at the top of the order for the first time in the tournament, launched a couple of big sixes before he edged the headband-wearing Hamid Hassan behind on 37.
Bell remained unbeaten on 52 alongside James Taylor to ensure England won their first-ever one-day meeting against the World Cup debutants with 41 balls to spare.
Ravi Bopara and James Tredwell were selected for the first time at this World Cup as they replaced injured duo Chris Woakes (foot) and Moeen Ali (side).
Both all-rounders appear to be doubtful for the upcoming Test tour of the Caribbean, with the squad to be named early this week, as Woakes arrived at the SCG on crutches and with his left foot in a protective boot.
That could be worrying news for coach Peter Moores who has seen his position come under increasing scrutiny after England’s only wins over the past month have come against associates nations Scotland and Afghanistan.
That meant they at least equalled their worst World Cup performance, in terms of wins, after England beat only Holland and the United Arab Emirates in 1996 - although at that tournament that was good enough to a quarter-final place.
The 15-run defeat to Bangladesh in Adelaide on Monday meant that England would not go to the last-eight this time around and England and Wales Cricket Board managing director Paul Downton has already confirmed a “major review” would be launched into their failings.
That could yet result in a clear-out of the one-day side, although veteran players such as James Anderson have already confirmed their desire to remain a part of England’s 50-over plans.
Anderson helped ensure that once skipper Eoin Morgan won the toss and sent Afghanistan in that England were quickly on top and only ever troubled by the weather.
Anderson and new-ball partner Stuart Broad both struck before the rain first held the game up after 8.5 overs.
Joe Root held both catches to remove openers Nawroz Mangal, who slashed at a wide ball, and Javed Ahmadi who offered a more routine chance.
Jordan struck twice after the restart as Afghanistan lost four for 17 around the break.
Afsar Zazai nicked through to wicketkeeper Jos Buttler before Morgan dived forward at point to catch Sami Shenwari.
Jordan almost had another only for Nasir Jamal to correctly review a catch behind after he was tucked up by a short ball. Jordan’s impressive spell read 6-2-9-2 before another rain delayed reduced the match to 45 overs per side.
It hardly appeared England would need that amount of time as Bopara had Nasir edging behind four balls after the restart.
New man Mohammad Nabi gave the small section of excitable Afghan fans reason for cheer, when he drove a six in their direction off James Tredwell, but the captain then slapped Bopara to the spinner at mid-on.
Bopara then showed his athleticism in the field to run around and take a diving catch at deep square-leg when top-scorer Shafiquallah Shafiq top-edged a sweep at Tredwell.
The rain returned for a third time and, while England were made to wait for their chance to reply, they made no mistake when the weather relented to at least ensure a winning end to their World Cup.