After slipping to a home defeat to Crystal Palace on Saturday, Solskjaer made 10 changes in Bedfordshire, although his starting line-up still boasted eight senior internationals.
Donny Van De Beek made his full debut in midfield, while goalkeeper Dean Henderson finally made his Manchester United bow, with Rashford required off the bench to sew up the victory.
A trip to Preston or Brighton awaits in round four, but Solskjaer knows his team need to find the form from the back end of last season if they are to remain in the hunt for the long haul this time around.
James Shea was unsurprisingly the busier of the goalkeepers in the opening exchanges and did well to keep out a Mata strike as the visitors saw plenty of the ball.
The lack of incisiveness from his side would have been a worry to Solskjaer, who had highlighted a late start to pre-season training as a key reason in their weekend loss to Palace.
But, just as Luton appeared set to have seen out the first half without incident, they conceded a penalty less than two minutes before the break.
A United attack broke down and Brandon Williams collected the ball, bursting into the box where he was tripped by George Moncur.
Mata stepped up to send Shea the wrong way and give the visitors a half-time lead.
The visitors started to exert a period of pressure as Jesse Lingard saw a shot cleared off the line before Shea once again did well to push clear a Nemanja Matic drive and then keep out Van De Beek with his legs.
Solskjaer clearly felt he needed more to put a spirited Luton to bed and introduced Bruno Fernandes, Greenwood and Rashford from the bench with 12 minutes to go.
Instead, it was Luton who would threaten next, Henderson making his first save of the night to deny Tom Lockyer, who then saw his follow-up effort cleared off the line.
But then the resources available to Solskjaer would prove the difference as Rashford and Greenwood both opened their accounts for the season in the dying stages to seal a result which certainly flattered the visitors.