The 29-year-old scored twice either side of a Gareth Bale goal as Andre Villas-Boas' side recorded their first three points of the campaign.
The visitors were impressive throughout the match and Defoe hopes this can be the result which gets their stuttering start to the season out of their system.
"Hopefully the fans are happy and everyone who has doubted us can look at us and think we'll do well this season," he told Sky Sports.
Defoe admitted that Spurs might have panicked if they had needed to defend a one-goal lead late on and therefore acknowledged that Gareth Bale's strike, which made it 2-0, was a welcome relief.
"That's what we wanted, just to settle us down," he said. "When we got the second goal, we passed it a lot more and [we] could control the game.
"When it's just 1-0, we panic a little bit because obviously we've conceded late goals. It's something we wanted to get out of our game really."
Their had been much speculation over Villas-Boas' future, with reports even suggesting that he had just three games to save his job, but Defoe dismissed that claim and says his new boss needs time to get his ideas across.
"I think that's normal to be honest [for there to be pressure on Villas-Boas]," he continued. "When a manager comes to a new club sometimes it takes time.
"You get world class players that come to football clubs and people think instantly they are going to be unbelievable sometimes.
"Hopefully the fans are happy and everyone who has doubted us can look at us and think we'll do well this season."
The England striker also insisted that he was happy to continue as a lone striker and felt that he could thrive in such a role if he received enough support.
"Yeah, it's good," he commented. "I've always said that when you're playing with quality players it's not a problem [playing alone] because they get close to me.
"Today we had a lot of the ball, so it's just a case of being clever and making sure your movement is good, and I did that today."