Kane gave his side the lead early as he found the bottom left corner from just outside the box in the 5th minute, before finishing off the game in the closing stages to continue Tottenham's undefeated start to the season.
Forest conceded a penalty in the second half through a handball from defender Steve Cook, but this time Henderson denied Kane, giving the keeper successive penalty saves at home after denying West Ham's Declan Rice two weeks ago.
Club-record signing Morgan Gibbs-White impressed on his debut for Forest, but the home side could not find a way back into the match as Tottenham moved to third in the table, while Forest find themselves in 12th.
Difference-maker Kane said after the match that despite the win, he felt that his side did not play their best on the day.
"We didn't play our best, we know that. We dug in and I think there's been a few games like that this season. We got the three points but for sure we know we can do better," Kane said.
"Any Premier League game we expect a tough match, especially away from home. We had a great preseason and physically feel really good. We're just not quite showing it yet. We're still winning games which is the most important thing. We're becoming stronger as the game goes on. There's a long season left and we're still grinding out wins when we're not playing our best which is important.
"We have a fantastic manager, a really strong squad and we will need that squad over the next period as it's a busy schedule. It's important those players who haven't started games yet to be ready for their opportunity. We have started well but as we know in the Premier League it can soon turn around."
Manager Antonio Conte said he was pleased with the result, but echoed Kane in saying that his team can do more.
"Our start [to the season] was good. It's important in the Premier League to try to get three points in every game. We have to continue in this way, it's not easy, another clean sheet and we scored two goals," Conte said.
"Many players are improving in many aspects. We have to trust in ourselves. In the Premier League there are top teams who want to try to fight.
"It's always a sporting war. After a game there's a winner and a loser. Every game has to be a sporting war for us."
Tottenham will face West Ham on Wednesday evening as they look to continue their run of form.