The 26-year-old striker came on as a substitute in the 79th minute and scored the only goal of the game on 87 minutes.
Redknapp, who worked with Austin at QPR, has taken to Twitter to express his delight at the former Burnley star’s performance against United.
I spoke to his grandad Ozzy last week. He follows him everywhere but wasn’t going on Saturday as he didn’t think Charlie was in the squad!— Harry Redknapp (@Redknapp) January 25, 2016
Arsenal legend Ian Wright was also impressed with Austin’s heroics against the Red Devils, and has suggested that Louis van Gaal’s side should have moved for him in the January transfer window.
“How the Manchester United manager could do with someone who finds the net with such regularity as him,” the former Arsenal striker has written in The Sun. “Eleven home games without a goal by half-time? That’s ridiculous.
"You certainly don’t struggle for them when Austin is around. He’s averaged better than a goal every other game all his career, all the way from Kintbury to the top flight.”
Southampton are eighth in the Premier League table at the moment with 33 points from 23 matches.
Ronald Koeman’s side are nine points behind Tottenham Hotspur, who are currently fourth and occupy the final Champions League spot.