England lasted only eight days in the 2014 World Cup, losing to Italy and Uruguay and drawing with Costa Rica in group play.
Gareth Southgate's men appeared to run out of ideas in the second half, with Tunisia seemingly set to see out what would have been an unexpected draw.
Rooney is England's all-time scorer in worldwide matches, scoring 53 goals in 119 matches.
Before he went off he made one more stunning block, denying John Stone's 11th minute header, but was helpless as Kane turned in the rebound for his first goal at a major finals.
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Two monitoring stations in Mexico City picked up the tremor as Hirving Lozano scored the game-winner 35 minutes into the match. Lozano, a 22-year-old forward nicknamed "Chucky", got Mexico going with a deflected shot over the bar in the first minute.
Tunisia suffered a further setback in the 16 minute when goalkeeper Mouez Hassan had to leave the field injured and was replaced by Farouk Ben Mustapha. Vice-captain Wahbi Khazri leads the squad with 12 career goals for the national team, while Saber Khalifa, Anice Badri and Fakhreddine Ben Youssef will lead Tunisia up front.
But Kane popped up in the 91s minute to head England to victory from a corner.
As half-time approached England worked a clever free-kick situation with the ball falling to Lingard unmarked, but he could only hit his shot against a defender.
Marcus Rashford replaced the ineffectual Raheem Sterling with 20 minutes remaining but still England struggled to create.
"I definitely didn't expect the manager to drop him against Slovenia and as soon as we all saw that we knew that that gaffer was not somebody to be messed around with", said Rose.
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At 30-years-old, it remains a question as to whether or not Kalinic will ever don a national jersey again. Croatia are back in action against Argentina in Nizhny Novgorod on Thursday.
The winning goal came at the end of a drab second half in which Tunisia defended deep and nullified England's attack.
Belgium will have considered themselves favourite for Group G's top spot coming into the competition, and their confidence won't have been dented after watching England get their first taste of action in Russian Federation.
Kane - who else? - popped up to break Tunisian hearts and the reaction of those in the BBC Sport studio was priceless.
"I'm so proud of the lads", Kane said.
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Hadary, the captain, said making it to the tournament was the culmination of 25 years of hard work. In terms of him on the pitch, he'll have full guarantees. "Certainly he doesn't feel afraid".