However, Tunisia drew level from the penalty spot 10 minutes before half-time when Ferjani Sassi converted from 12 yards. Pictured centre: Kane is mobbed by team-mates after his opener; top right: fans celebrate in Volgograd; top left: Rebekah Vardy watches the action.
England lasted only eight days in the 2014 World Cup, losing to Italy and Uruguay and drawing with Costa Rica in group play.
Tunisia mounted a bit of a comeback after that and were awarded a penalty in the 33 minute when Kyle Walker brought Fakhreddine Ben Youssef down inside the box.
Tunisia were the challenge that lay ahead of England as they got their World Cup campaign underway on Monday night.
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"I think we deserved the win", he said.
'We've got this result over the past few weeks and that's because of everyone.
Tunisia will be playing in the World Cup for the first time since 2006 and seeking its first victory since 1978. We had to keep creating chances, we didn't play as well second half but they defended deep. "We kept our shape and "H" has done it again", he said, referring to Kane's double strike.
England continued to press and were made to pay for not converting a succession of chances when they conceded a soft penalty.
Belgium sent a statement to their Group G rivals after they thumped Panama 3-0 in the earlier kick-off to move to the summit of the pool.
Kane had given England the lead in the 11th minute after a bright start.