The Red Devils headed into a home date with the Saints on Saturday boasting an unbeaten Premier League record under their interim boss.
At times it appeared as though Solskjaer could suffer just the second setback of his reign at Old Trafford, with the visitors taking the lead and putting in a spirited showing.
United, though, have rediscovered their fighting spirit in the wake of Jose Mourinho’s departure and dug deep to claim a priceless three points which has lifted them back into the top four.
Solskjaer told BBC Sport: “It was one of those games, it had everything.
“It was an open game and they came here to beat us. We should have been 1-0, 2-0 up in the first five minutes.
“They settled down and scored with two unbelievable strikes. My team did fantastic and the boys did great in the second half.
“The dressing room is bouncing at the moment.”
Lukaku, who started alongside the fit-again Marcus Rashford, netted a match-winning double for United.
Andreas Pereira also recorded his first Premier League goal as the buoyant Red Devils left it late but clawed their way over the line – with Paul Pogba even able to miss a stoppage-time penalty.
“The belief is there, they have gone so long without losing games [domestically] and are raring to go,” Solskjaer said of a side that are 12 games without defeat in the English top flight.
“They want to get on the ball and the confidence is high. They were class finishes from Romelu Lukaku and I am so pleased for Andreas Pereira.
“Rom is a goalscorer, he keeps practising every single day and has quality.
“When we changed to two up front that helped him as well. We had some great play at times, 3-2 we did it the hard way. That's why we are turning grey.”
United will need to find an attacking spark again in their next outing, as attention switches to European competition.
Solskjaer has already conceded that the Red Devils have a mountain to climb in order to reach the quarter-finals, but claims confidence is high heading into the toughest of continental tests.
He said: “We have done fantastically well away from home but we go to PSG with the belief, we have nothing to lose.
“We will give it a go. This club has done so many great comebacks, the tie is not done.”