The two-time winners of the tournament have looked unconvincing in their three matches - edging Associate sides Nepal and Italy while also losing to West Indies.
England opted to bat first at the toss but then slipped to 67-3 in the eighth over with Phil Salt (28), Jos Buttler (3) and Jacob Bethell (23) all back in the dressing room.
The trend of getting starts and then getting continued Tom Banton struck 30 while Sam Curran scored 25 but England found themselves at 159-6 in the 17th overs.
It took a brilliant cameo of 53 not out off 28 balls from Will Jacks to lift England to a total of 202-7.
Jofra Archer then rocked the Italian run-chase with two wickets in the first over to leave the underdogs at 1-2 with Anthony Mosca and Jon-Jon Smuts failing to trouble the scorers.
Ben Manenti though kept Italy in the contest with a destructive 90 off 25 balls while Grant Stewart pummeled 45 from 23 deliveries to keep the upset alive.
Jacks then removed Manenti while Curran took the scalp of Stewart in the penultimate over as England eventually completed the victory.
Yuvraj Samra hit a record-breaking century for Canada but New Zealand aced the run-chase to come away with an eight-wicket win to secure their place in the T20 World Cup Super 8s.
A sensational unbeaten century from Pathum Nissanka powered Sri Lanka to a superb eight-wicket victory over Australia in T20 World Cup action on Monday.
Azmatullah Omarzai starred with bat and ball to help Afghanistan register their first win of their T20 World Cup campaign after beating UAE by five wickets with four balls to spare.
Australia have confirmed that Steve Smith has officially been added to their T20 World Cup squad, replacing injured fast bowler Josh Hazlewood.
