Caretaker boss Solskjaer will be without nine crocked first-team players, while £89m midfielder Paul Pogba is suspended after his red card late on in that first leg defeat.
Ever-improving James Ward-Prowse bent home a brilliant free-kick to hush Old Trafford, but Lukaku had the final say by lashing home to seal a 3-2 win that propels Solskjaer's side back into the top four.
The side went down early in the first half when De Gea was beaten by a long-range shot.
The 30-year-old has only managed one Premier League goal this season and is now facing "six to eight weeks" on the sidelines with the knee injury that he picked up against Southampton on Saturday afternoon.
A brace from Romelu Lukaku and one from Ander Herrera handed the Red Devils maximum points after regulation period. Credit was still due to the striker for knowing where the goal was and shooting nearly immediately, and though Gunn was beaten for a third time, he did manage to keep out Pogba's stoppage-time penalty, awarded after Armstrong had fouled Rashford.
Manchester United 3 Southampton 2: Lukaku seals thrilling comeback
And Solskjaer, who famously scored the injury-time victor in the 1999 Champions League final, felt there was an element of nostalgia to United's triumph.
Marcus Rashford headed just wide when reaching the substitute's cross, before the Portuguese provided an assist for a goal that Pereira will rightly take the plaudits for. If he misses he misses, can he improve his penalties?
It's special because we came along together and we all know each other well. So I think he probably has to learn from that. We spoke about it at half-time and said get the ball in the box and play more forward because the Stretford End will suck it in for you. "And he goes happy again, and that for me is the most important thing".
Solskjaer's replies were by virtue of his performances, as he went on to collect 26 points from his first ten games in charge, the highest-ever by any Premier League manager.
Sergio Aguero scored the only goal of the game from the penalty spot to ensure that City remain within a point of Liverpool, who thumped Watford 5-0 on the same evening to send out a message.
Jose Mourinho has said that he doesn't believe that Ole Gunnar Solskjaer will be able to maintain his start to life at Manchester United for the long-term.
To their credit, Ralph Hasenhuttl's Saints side was not done there, and they were level with a free kick from James Ward-Prowse, before the Belgium global won the match for United late on.