United suffered a 3-1 home loss to Crystal Palace in their first Premier League game on Saturday and Solskjaer made 10 changes for the third-round tie at Kenilworth Road, including handing a debut to goalkeeper Henderson as David de Gea was given the night off.
A Juan Mata penalty on the stroke of half-time gave United the lead against its Championship host, which would have levelled with nine minutes remaining had Henderson's one-handed stop not denied Tom Lockyer.
By that point a concerned Solskjaer had called for the cavalry and substitutes Marcus Rashford and Mason Greenwood both added late goals for a victory that flattered United.
Harry Maguire was the only United player retained from those who started the defeat to Palace, though the newcomers looked just as rusty as those they had replaced.
On one of the few moments United's attack clicked, James Shea saved from Mata after Donny van de Beek stepped over Jesse Lingard's pullback.
Luton, which hit the side netting through Danny Hylton, looked like entering the break on level terms but George Moncur's clumsy barge on Brandon Williams resulted in a 44th-minute penalty that Mata tucked away.
There was little improvement from the visitor after the break, though Lingard did have an effort cleared off the line by Ryan Tunnicliffe, while Shea was equal to efforts from Nemanja Matic and Van de Beek.
Solskjaer brought on attackers Bruno Fernandes, Rashford and Greenwood with 11 minutes to go, though the next goal very nearly came at the other end, with Henderson springing to his right to push out Lockyer's header before Eric Bailly blocked the follow-up attempt.
Rashford collected Greenwood's pass and slotted home to kill off any Luton hopes with two minutes remaining, before the teenager opened his account for the campaign with United's third in stoppage time.