Monday's victory at Eden Gardens in Kolkata meant England finished second in group C, behind West Indies who also qualify, and now go to Sri Lanka for the next stage.
England will need to improve markedly if they are to reach the last four, but after a chastening Ashes loss they will be relieved at avoiding an embarrassing loss.
Italy, in their debut tournament, worried England with bat and ball.
Opting to bat first England were reduced to 5-105 in the 13th over before Jacks (53no off 22 balls) added 97 off the last 44 balls enjoying good partnerships with Surrey teammates Sam Curran (25 off 19) and Jamie Overton (15 off nine).
That took England to 7-202, which looked like being enough against a team of mostly part-time cricketers whose highest chase is 155 against Ireland in a warm-up last month.
It was enough, but only just with the 24-run margin of victory not reflecting the closeness of the game.
Sydney Sixers' allrounder Ben Manenti, grade cricketer Justin Mosca, a Sydney PE teacher, and Kent county pro Grant Stewart gave England an almighty scare.
After Jofra Archer (2-35) had taken two wickets in the opening over and Overton (3-18) dismissed Harry Manenti (12), the skipper's elder brother Ben (60 off 25) and Mosca (43 off 34) put on 92 in eight overs.
Ben Manenti's departure looked to spell the end but Stewart (45 off 23) took the game deep, hitting both Archer and Adil Rashid for two sixes in an over. However, when he perished to Sam Curran (3-22) England could at last relax.
"The most important thing is we've qualified," said Jacks. "It doesn't matter how you do that but we know we need to be a lot better."
But there are question marks over England. Openers Phil Salt and Jos Buttler are struggling to put together partnerships with the latter averaging 13.25 at the tournament so far. No one has made a score above 63. The bowling has been inconsistent, but will prefer Sri Lankan surfaces.
A series of England batters got in, then out, with only Tom Banton reaching 30 until Jacks cut loose.
