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The 21-year-old arrived at Old Trafford from Borussia Dortmund in the summer and is yet to fully settle in.
He started on the bench of Sunday's 2-1 win over West Ham United and Keane, who is known for his fiery temper, spoke sense when it came to Sancho.
"He'll be fine," the former United captain told Sky Sports per the Express.
"He's been at the club what, two months? There's been a lot of changes, new players coming to the club - give the kid a break.
"He's still learning his trade, he's still a young player, big expectations.
"He's come to a new club where he's certainly not going to be star man, he'll be down the pecking order and he's got to adapt to all that. Give him a chance."
Sancho had been on United's radar for 18 months and his signing brought much fanfare.
However, the unexpected arrivals of Raphael Varane and Cristiano Ronaldo bolstered Ole Gunnar Solskjaer's side even further.
One position United failed to invest in is the central midfield department and Keane thinks it'll cause problems this season.
West Ham lined up with Declan Rice in midfield at the London Stadium and he's a long-term target.
When asked if the 22-year-old could improve United, Keane replied: "I think he would, I think he's improved as much as any player over the last one or two years that I've looked at him.
"I know he's been linked with United and if you had asked me a year ago, I'd have said: 'No, let's give him a year or two [before he's ready for the transfer],' in terms of maturing.
I look at him and see he's 22 - he's way ahead of where I was at 22.
"I think the next step will come for him when he's playing at a higher level. When he's playing for Man United, playing in the Champions League, competing for trophies.
"He's obviously a regular for England. There are a lot of plusses for him. There's a lot more to come from him.
"He's a big presence out there. I think it will be a really good challenge for him today because I'm not a big fan of the United midfield. I think they're a bit short, so we will see how he does today against them."
Said Benrahma gave David Moyes' side the lead inside half an hour after his effort deflected in off Raphael Varane.
Ronaldo drew United level five minutes later with a close-range effort.
That'd be it for goalmouth action until the 89th minute when Jesse Lingard, who spent the second half of last season on loan at West Ham, fired in a winner.
More drama followed after Luke Shaw gave away a penalty in the dying moments with a handball in the area.
West Ham brought on Mark Noble to take the spot-kick and it was remarkably saved by David de Gea, his first penalty stop in five years.