Record Jacks fifty carries England to 202-7 in must-win Italy match
Most of England's batsmen got good starts against the lowest-ranked team in the tournament after Harry Brook won the toss and chose to bat.
But only Jacks, who reached his fifty off 21 balls, capitalised as Italy kept chipping away with regular wickets.
Jacks took the game by the scruff of the neck after England stuttered to 105-5 in the 13th over.
He put on 54 in 25 balls with Sam Curran (25) for the sixth wicket and 35 in 16 balls with Jamie Overton (15) for the seventh.
His fourth six, which took him to his fifty, also saw England pass 200 for the first time in the tournament, with Jacks finishing 53 not out off 22 balls.
An England win will guarantee their passage to the Super Eights along with the already-qualified West Indies from Group C.
A defeat will leave them with two wins and two losses from their four matches and reliant on the final group results this week going their way.
England named an unchanged side for the third match running and were 56-2 at the end of the six-over power play, having lost both openers.
Jos Buttler, who has failed to pass 30 in the tournament so far, continued his run of low scores with three.
Phil Salt got off to a rapid start before also falling for 28 off 15 balls with two sixes.
Both were caught attempting big hits and, when Jacob Bethell followed suit for 23, England were wobbling at 67-3 in the eighth over.
Brook took a four and a six off the first two balls of JJ Smuts' third over.
He edged the next ball to wicketkeeper Gian-Piero Meade trying to cut and England were 92-4 after 11 overs.