Greenwood's brilliant goal gave United a 1-0 win over Astana at Old Trafford on Thursday.
And Hargreaves saw glimpses of former United striker van Persie in the 17-year-old.
He said: "I think you have to take the positives on a night like this and the positives are that they kept a clean sheet and Mason Greenwood scored a stunning goal.
"It reminded me a little bit of Robin van Persie, chopping inside onto that right foot and scoring a great goal.
"I'm really happy for him because, as you saw (Bukayo) Saka for Arsenal, the night was all about the academies really.