Headers from Harry Maguire and Dele Alli either side of half-time saw England ease to a 2-0 triumph in Samara.
Kane was kept relatively quiet, having scored in six consecutive international appearances, but becomes the first England captain to lead the Three Lions into a FIFA World Cup semi-final since 1990.
"So much preparation for this game, a quick recovery from the last round [a penalty shootout defeat of Colombia]," Kane said to BBC Sport.
"We were fantastic. Sweden made it tough, it was a difficult game. Long balls and different to what we have faced so far. We were brilliant on set-pieces, crosses into the box, we're buzzing.
"It's a big game ahead. We feel good. Confident. We looked composed, we controlled the game.
"The fans here enjoy it, I'm sure the fans back at home - we'll see videos of them enjoying it. We'll keep doing what we're doing to make the country proud."
Maguire's first international goal put England on track, the Leicester City defender rising highest in the penalty area to crash home a header from Ashley Young's corner.
"Can't do better than that, [scoring in the] quarter-final of the World Cup," Maguire told BBC Sport. "Winning is the main thing and we've done that.
"We felt we could dominate the ball and control the play although we got a bit sloppy near the end of the second half. We knew what would hurt them and what we'd worked on.
"Set-pieces is a big thing, and that cross from [Jesse] Lingard to Dele, so all credit to the players for executing it."
