Kane put Spurs in front on 25 minutes with a close-range finish to record his 25th goal in all competitions this season, his best tally since scoring 41 for Tottenham in 2017-18.
Tottenham doubled their lead through Kane as the second half wore on after the England international pounced on a tame clearance and beat Zagreb keeper Dominik Livakovic with a low, driven shot for his 26th of the campaign.
The only worry for Tottenham, who face Arsenal in the north London derby on Sunday, was the sight of Kane having an ice pack applied to his knee after he was substituted late on.
Spurs manager Jose Mourinho made seven changes to the side that beat Crystal Palace in the Premier League last weekend, but there was never any doubt that talisman Kane would start.
After the match, Mourinho weighed in on the knock to Kane: "Harry, I don't believe is anything big. Sunday is a match everyone wants to play and little things will not stop him. I don't have the feeling it is anything serious.
"He is doing everything. He is doing the striker job of scoring goals. the team work of creating space and linking play and the extra work not many strikers do in defence. He helps us a lot. He has an incredible number of clearances from crosses and corners.
"We couldn't be happier. I gave a rest to many of the players and am so happy with what Vinicius is giving us. But with Harry on the pitch it is a different story for us."
The Tottenham manager has made no secret of his desire to win the competition, a feat he managed with Manchester United in 2017, and with Kane so prolific they have every chance.
"He is just enjoying his football. As we can see he attracts the ball," Spurs keeper and captain Hugo Lloris said.
"[Erik] Lamela made an amazing job for the first goal and Harry was there at the right time to score. His second gives some relief to the team before the second leg."
Tottenham could make little impression in the early stages, but inevitably it was Kane who broke the deadlock, although it was Lamela who was the instigator.
The Argentine jinked his way into the area before clipping a shot with the outside of his right foot against the post, the ball rebounding to Kane who made no mistake.
Dinamo showed enough to suggest they could be a handful in the second leg, not least when Bruno Petkovic wasted a great chance on the hour when he shot wide.
Mourinho made a triple substitution with Gareth Bale, Steven Bergwijn and Lucas Moura all coming on.
Bale made an immediate impact with a glorious cross to Bergwijn whose volley was saved by Dominik Livakovic and also forced Livakovic into a save from a thunderous free kick from 30 metres out.
The teams will meet again next Thursday for the decisive second leg in Croatia.