Lucas Moura pulled one back for Spurs, but it was not enough to stop West Ham claiming the win that saw them leapfrog Chelsea into fourth.
Rice and his team-mates are now dreaming of qualifying for next season’s Champions League, but he believes the next four games will be crucial.
West Ham travel to Premier League leaders Manchester City on Saturday and then they face Leeds, Manchester United and Arsenal before next month’s international break.
“I think I would be wrong to sit here and say that’s not something that I am thinking about,” Rice told talkSPORT.
“I want to be in those [Champions League] places now. This club, this is where it deserves to be, it’s a massive club. We have got quality and that’s where we need to be pushing.
“I think the next four games for us are going to define that. We have got City, then Leeds, then United and then Arsenal. They’re four massive games.
“If we get through those with a good amount of points, who knows. We will be down to single figures [of games left] then and we will be able to start counting down to see how much longer we’ve got to go.
“I can say first hand probably this is the best West Ham team that I have been apart of in the four years that I have been in the team.
“The mentality has just changed completely and I think you can see that on the pitch. We are fighting for each other.”