Arsenal beat West Brom 3-2 at the Hawthorns to end the Barclays Premier League campaign one point ahead of Spurs, who recorded what was in the end a hollow victory over Fulham at White Hart Lane, although fourth place could yet be good enough for a place in next season's Champions League qualifiers.
It completed a remarkable turnaround for the Gunners, who slumped to 17th after losing four of their opening seven league games and at one time trailed Spurs by 10 points.
Van Persie told Arsenal Player: "It was double important to finish third and also above Spurs. We knew what was at stake.
"It is just a fantastic feeling again to be ahead of Spurs. We heard that they won, but it doesn't really matter because we won our game.
"We are definitely in the Champions League again, and that was the main target from the match.
"We just had to qualify and everyone is very pleased, everyone is enjoying this moment."
Van Persie praised the battling efforts of Arsene Wenger's squad during what has been a testing season.
"It is incredible. We had to dig deep and come back on a couple of occasions where we were having a hard time," the Dutchman said.
"At the start of the season, we had a very difficult period, but we bounced back very well from that.
"After that we had the games against AC Milan (in the Champions League) and Sunderland (in the FA Cup) which were quite a hard blow, but we did bounce back."
Van Persie netted 30 Premier League goals to help fire the Gunners back into the top four.
The Holland forward spent plenty of time applauding travelling supporters after the final whistle at The Hawthorns, and is set for talks with Arsenal over his future this week.
Should he remain at Arsenal and see out his contract next season, Van Persie could well be paired with Lukas Podolski, who is set to arrive from Cologne in a £10million move.
The German striker believes the switch to England has come just at the right time.
Podolski said: "For me it does not really matter when you take such a step in your career, whether it is now or a few years earlier.
"It is difficult to say when the time is right, but I am now 26, and nearly 27, so this step in my career might just be the right time to do it.
"I had a very good season with Cologne, I played well and as a result I am now going to Euro 2012 in the summer with lots of confidence.
"After that, I will be joining up with my new team-mates at Arsenal and it is my goal just to do my best and be successful there."
Podolski is ready to slot into the Arsenal side.
"In Cologne, I played as a centre-forward and also as a playmaker. In the national team I play wide on the left, I honestly don't mind where I play, as long as it is in a forward position," he said.
"But it is always the coach who decides where you play and I will do my best wherever I play."