The 19-year-old, who joined from Monaco on transfer deadline day, scored twice in Sunday's 3-2 win over Southampton as United moved to second in the table.
The Frenchman now has three goals from his first two Premier League games.
"He's been absolutely magnificent. They could not have hoped for any more," former Arsenal and England defender Keown told Match of the Day 3.
"There was a huge transfer fee on his head, but it doesn't seem to bother him. Every time he is in front of goal he seems to be at his happiest."
Martial became the most expensive teenager ever when he moved to Old Trafford, and the signing was met with surprise by some commentators in the media.
Already an established France Under-21 international, he had just been called up to the senior national team when he joined Louis van Gaal's side at the beginning of September.
And with team-mate Wayne Rooney having yet to score in the league this season, Martial has already proved vital to United.
"I thought they would take him off at 3-1 and they do need to manage him well. He's only a young boy really," said Keown.
"But he looks very powerful and mature and he's well beyond the age he played at on Sunday."
Manchester United forward Anthony Martial has been unfazed by his "huge" £36m transfer fee, says Martin Keown.