England, who lost on penalties to Italy in the Euro 2020 final at the same stadium in 2021, equalized in the 32nd minute when Jude Bellingham sprinted into the area and was brought down by a late Di Lorenzo challenge.
After a lengthy VAR review, captain Kane calmly stepped up and tapped home, sending Gianluigi Donnarumma diving the wrong way.
Rashford made it 2-1 in the 57th minute with a scorching shot into the bottom corner after being set up by Real Madrid forward Bellingham.
Kane then scored his second goal of the night in the 77th minute, dribbling past players to score as manager Gareth Southgate punched the air in delight as his team qualified unbeaten and with 16 points from six games.
Kane now has 24 goals at Wembley, one more than 1966 World Cup champion Bobby Charlton.
The result left Luciano Spalletti’s Italy with 10 points and a duel with Ukraine for the group’s second qualifying spot.
The change was immediate, with neat passing and Bellingham pushing midfield, although Italy remained dangerous on the break.
On Friday, Jordan Henderson, the starting captain, came off the bench.
A moment of silence was observed before the game after a shooter killed two Swedish soccer fans on the eve of a qualifier against Belgium in Brussels on Monday.
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