Faced with an eight-point gap before kickoff at White Hart Lane, second-placed Tottenham knew they could not afford to slip up if they were to keep alive their hopes of being crowned champions of England for the first time since 1961.
Any apprehension disappeared after 44 seconds, though, as Kane converted Kyle Walker's cross and he struck again after 16 minutes, calmly slotting home after Dele Alli's astute pass.
He has 21 league goals, two more than Leicester's Jamie Vardy who he will join up with for England duty next week.
"I'd rather have the five-point lead, but with Leicester usually playing first we know what we have to do -- there's still a lot of points up for grabs," said Kane, the first Tottenham striker to score more than 20 league goals in consecutive seasons in 45 years.
Bournemouth, bidding for a fourth straight league win to all but secure a second top-flight season, made little impression and Tottenham extended their lead seven minutes after the break when keeper Artur Boruc parried Kane's shot and Christian Eriksen was on hand to convert.
"The performance was outstanding," Spurs manager Mauricio Pochettino told the club's website, picking out Kane for praise.
"He's a fighter, his skills are great, he's a Tottenham supporter. We are lucky to have him."
Tottenham face back-to-back games away to Liverpool and at home to Manchester United after the international break, which Pochettino said has come at a good time.
"It's important to relax a little and analyse," he said.
"Ninety percent of the squad are with their national teams. After that we have seven games left and I hope they come back with a good sprit to fight."
Bournemouth manager Eddie Howe was disappointed with his side's display, but said Tottenham could go and win the title.
"No disrespect to anyone else but in our eyes they are the best team we have played," he said.
(Reporting by Martyn Herman; editing by Ken Ferris and Brian Homewood)
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