Van Persie followed Wayne Rooney's 40-yard pass, watching it drop 20 yards from goal before drilling a magnificent volley straight past Villa goalkeeper Brad Guzan.
It was the second of Van Persie's hat-trick, which sealed a 3-0 win, cemented a 20th league championship for the Red Devils and swept the Dutchman to the top of the Golden Boot standings with 24 - one more than nearest challenger Luis Suarez, who seems unlikely to play again this season.
"It has to be right up there," said Carrick of Van Persie's volley.
"The movement for starters, then Wayne to pick him out with a great pass. The finish was textbook.
"There are not many people who can do that.
"Robin has had a terrific season. It is his first one here and if he carries on like this he will have a good career that's for certain."
Carrick also reserved words of praise for 39-year-old Ryan Giggs.
Like his manager, Sir Alex Ferguson, Giggs has now won 13 titles on his own, as many as Arsenal have collected as a club.
"I keep talking about him but what else is there left to say," said Carrick.
"It is sensational that he is still able to do that after all this time.
"It is not just physical, it is mental, having the drive and desire to do it every day.
"The league wasn't won tonight [Monday], it was won all season with all the work we have put in.
"Ryan is out there training every day. The standard he has set is unbelievable.
"He is a legend. It is a throwaway comment too often. But he is the legend really. It is a pleasure to play with him and work with him every day."