The 29-year-old scored a hat-trick in United's 3-0 win over Aston Villa to ensure that the Premier League title came back to Old Trafford this season.
The former Arsenal man expressed his delight, claiming that it was a "massive relief", but also admitted that he is now focused on winning more silverware with United as well as the World Cup with the Netherlands.
"I've waited a very long time for this — it's what every professional dreams might happen one day," the Dutch striker told reporters. "That's why I sacrificed so much to come here. I chased a dream and I'm so happy the dream has now been fulfilled.
"I still can't believe it. It's just an incredible feeling. It is also a massive relief — after all this time, to be lucky enough to win a title in one of the hardest leagues in the world.
"Quite simply, I now want to win as much as possible. Now I've won one big prize I'm even more eager to win more, even more eager to win it all. Next season I have the Premier League, FA Cup, Champions League, Capital One Cup, Community Shield and of course the World Cup to focus on."
Van Persie is the current Premier League top goalscorer, having hit 24 goals so far this season.