United were beaten 2-1 at home by Norwich City on Saturday in yet another lacklustre display.
Van Gaal has been criticised by some supporters for his side's style of play, and the defeat to the Canaries meant United have gone six games in all competitions without victory for the first time since 1998.
It leaves the Reds outside the Champions League places on goal difference and speculation that former Chelsea boss Jose Mourinho could replace van Gaal in the Old Trafford has been rife.
'Let him see it through'
But speaking on BBC's Match of the Day, Keown said it would be a mistake to get rid of the Dutchman.
"Since he went out of the Champions League I think it's really cranked up the pressure," Keown said on MOTD.
"Giggs was there (animated on the touchline) auditioning for the job almost.
"Mourinho now, top manager, looking around for a job it does put pressure but I think they should stick with van Gaal.
"Let him see it through but if they don't win a trophy this year that's three years at Manchester United without winning anything, which is a disaster."
Mourinho antics could see Red Devils follow Keown's advice
It seems strange that there would be any question marks over a manager of Mourinho's calibre taking over a struggling Manchester United.
But that reluctance, certainly on behalf of a large section of the Red Devils' support, comes from a wariness over the former Real Madrid manager's style of football and antics.
According to the Guardian, United want assurances from Mourinho that he understands the club’s traditions and is willing to fall into line if they decide that the former Chelsea manager should take over.
The paper reports that there are concerns at the highest level of Old Trafford about Mourinho’s managerial style and specifically the elements of his work that led them to decide against employing him when Sir Alex Ferguson retired in 2013.
So Keown's advice to let van Gaal 'see it through' could prove the best course of action until the end of the season.