United have endured a tough summer, making just one signing, and lost their Premier League opener against Crystal Palace at the weekend.
They picked up their first victory of the season at Kenilworth Road with Juan Mata scoring a penalty before late strikes from Marcus Rashford and Mason Greenwood.
It meant Solskjaer became the third quickest United boss to 50 wins with it taking the Norwegian 92 competitive matches.
Ernest Mangnall took 78 games while Jose Mourinho needed three more, while Sir Alex Ferguson did not reach the tally until 104 matches.
Though Solskjaer, who rested a number of players, was not caring about that, and was just happy to progress into the next round with United facing Brighton or Preston next week.
He said: "Well, going through is the big thing, there are some positives.
“The keeper [Dean Henderson] does his job, we scored some nice goals towards the end, apart from that it was a good workout.
“We just have to take it day by day and game by game. I didn't want to give [substitutes Bruno Fernandes, Marcus Rashford and Mason Greenwood] more than that because we will have a hard day tomorrow."
Paul Pogba, Anthony Martial and David de Gea were not in the squad with a first Manchester United start for Dean Henderson, who had little to do throughout the 90 minutes against the Championship side.
Mata put United ahead late on in the first half after Luke Shaw was felled in the box.
Henderson did pull off a strong save as he palmed away Tom Lockyer’s header as Luton pushed on in the second half.
Though the introduction of Bruno Fernandes, Rashford and Greenwood allowed United to pick off the Hatters’ tired defence.
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Rashford finished smartly past James Shea in the 88th minute before Greenwood rolled home in stoppage time with the final result perhaps a tad more than United merited.
Goalscorer Juan Mata added: "It is one of those games where if you win you did what you had to do and if you lose it's a mess.
“We did a professional job today, we didn't concede a goal, we scored three goals and we played some good football.
“It was important to win after our first game in the Premier League which we know wasn't good enough."