In a game of spectacular goals, Saints midfielder James Ward-Prowse struck a superb 25-yard free-kick after 74 minutes to make the score 2-2.
"We maybe could have had two penalties and at 2-0 up it would be much more hard for Man United", Southampton manager Ralph Hasenhuettl said.
Liverpool have a one-point lead over holders and defending champions Manchester City with ten league games left to play this season.
United are unbeaten in the Premier League since Solskjaer took over from Jose Mourinho in December, winning nine of their 11 games to revive hopes of a top four finish.
It looked like Lukaku was going to endure another frustrating afternoon as he missed two gilt-edged chances early on against Southampton, with fans venting their anger from the stands.
"I've done a photo shoot or a video shoot on season tickets, so I'm pleading for them to get their season tickets", he said with a smile.
Andres Pereira drifted through Southampton's defence to curl past Angus Gunn, before Romelu Lukaku engineered some space to fire home into the bottom corner on the hour mark. We lacked a bit of class, we should have been 2-0 up after five minutes.
And the Norwegian appeared to have summoned the spirits of the Ferguson era, after United claimed a crucial three points against lowly Southampton. "You want that character and today was one of those he blasted so he'll be OK".
But within 15 minutes of the restart, United were ahead.
"He has been on a good run but he has been an important player for us and will bounce back". My team did fantastic and the boys did great in the second half.
"He's got high energy quality in his passing, crossing and shooting, so pleased for him, a goal and an assist, it's a nice day for him to remember".
"We've got Arsenal coming up, PSG away coming up, the big games". "This club does great comebacks", the caretaker manager said.
It's hard to compare them, but Edwin and Peter were never player of the year, David's meant so many points for us over the years and I would say he's the best goalkeeper in the world, Peter was at the time and Edwin at the time as well, I'm delighted with David and his personality is fantastic.