Kane scored the third goal of his burgeoning international career to break Switzerland's stubborn resistance in the 67th minute before Rooney sealed a 2-0 victory, England's eighth win in as many Group E games.
The Tottenham striker was a second-half substitute and found the back of the net 10 minutes after coming on to earn his fourth England cap with a smart left-footed finish from Luke Shaw's low cross.
The international break has been productive for Kane, who came off the bench to score against San Marino, a 6-0 victory that booked England's ticket to next year's finals in France, and Switzerland.
The striker, who scored 31 times last season, is yet to open his Premier League account in four games so far but is confident of kick-starting his campaign when he returns to club football against Sunderland on Sunday.
"Any striker wants to be scoring goals, so it's great to come away and get a couple," the 22-year-old Kane told reporters.
"I'll go back and focus on Spurs but if I don't score at the weekend it's not the end of the world.
"Hopefully I can get some chances for the club now and start scoring goals and really kick on this season," he added.
Kane is also hopeful that he can one day overtake Rooney as England's top scorer.
"Rooney has set the mark and will probably go on to score quite a few more hopefully for us," Kane said.
"We'll see where he ends up and hopefully I'll be chasing him down. I'm sure he's very proud of what he has achieved."
(Reporting by Simon Jennings in Bengaluru; Editing by John O'Brien)
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