The £11.5million summer signing from Ajax was included in the starting line-up for the Premier League game against Norwich at White Hart Lane, and claimed an assist for Gylfi Sigurdsson's 28th-minute opener.
He also had a hand in the Icelander's second goal four minutes after the restart, before being replaced by Lewis Holtby.
Eriksen was appreciative of the warm reception he was afforded by the supporters, saying afterwards: "It felt really good. I'm really happy to be here."
When asked if the fans might see their team mount a title challenge, he replied: "Hopefully this season, or maybe after. But this is an exciting season.
"They can expect great things, they're allowed! We'll try to do our best."
For the opening goal, the Denmark midfielder played a neat pass into the area for Sigurdsson to run onto and score.
"I heard him yelling!" said Eriksen. "I also knew there was the defender there - that's why I didn't give it too much. Luckily, he was there in the right moment.
"I would say maybe we should have won by more, but it was probably fair. It's easier to perform when everyone around you is really good."
Eriksen turned down several offers from elsewhere before deciding to sign for Spurs, and he insists it was more about the allure of Tottenham than the prospect of playing in the Premier League.
"It always depends on the club. The club is the most important," he added. "Both felt really good, but I chose Tottenham."