Kane opened the scoring inside the first 11 minutes before Ferjani Sassi struck later in the first half from the penalty spot to draw level in Volgograd.
As the clock ticked by, it looked as though Gareth Southgate's men were destined for a draw, but Kane nabbed a second in the 91st minute to give his country all three points.
After the match, Rashford, who replaced Raheem Sterling in the 68th minute, told reporters: "He's been a brilliant leader for us leading up to the tournament, as well as now that we have started the tournament.
"He always leads by example and there's no better way to do it than by scoring goals, like he has been doing. He's a top forward. His career is only going to go up and up.
"He's very young and at this stage of his career, he's going to keep improving and getting better, so who knows how good he can get?"
Next up for England is a clash against Panama on Sunday.