The Red Devils appointed the former Manchester United striker as Jose Mourinho’s interim replacement last week.
The 20-time English champions opted to sack Mourinho following a 3-1 loss to bitter rivals Liverpool FC at Anfield last weekend.
Mourinho left Manchester United in sixth place in the Premier League table and a long way behind the top four.
Solskjaer has limited experience in the Premier League following a brief eight-month stint in charge of Cardiff City in 2014.
The retired Norwegian striker won six Premier League titles, the Champions League and three FA Cup crowns during his 11-season spell at Manchester United.
Former Manchester United captain Keane believes Solskjaer will boost the morale around the club and restore the feel-good factor within the Red Devils squad.
“The beauty for Ole is it is a case of getting these lads back on side and that includes Pogba and whatever has gone on in the past you have to move on,” Keane told BBC Sport.
“Ole will get that feel-good factor back, Mike Phelan back in is a good move, Ole will be popular around the club and I think getting the players back on side will be the easiest part of the job.
“They wanted a change and that is easy for Ole, you’d expect him to win the next few games, try and close the gap on the top teams, go on an FA Cup run and try and beat Paris St-Germain [in the Champions League].
“It is a big challenge ahead, but in the short term I expect the players to react and win the next few matches they should be winning.”
Keane added: “The feel-good factor will come back, but the problems are still there when they play the bigger games in the next few months. It is hard to cut the gap on Liverpool and City.
“But Ole will want to get the first game out of the way, a victory under his belt and no doubt a lot will be happy to see him back at the club, as he is such a decent guy.”
Keane was Solskjaer’s captain throughout the majority of his time at Manchester United as a player.
The Norwegian etched his name into Manchester United folklore with his winning goal in added time of their 2-1 victory over Bayern Munich in the Champions League final at Camp Nou in 1999.
The Red Devils finished in second place in the Premier league table last season, but Manchester United are a long way off the top-four pace in the current campaign.
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