United trailed 2-0 on aggregate after the first leg at Old Trafford last month and were left with an even bigger mountain to climb after losing 10 first team players for the return game in Paris.
But Lukaku scored twice - his third brace in his last three games - to give United a chance, before Marcus Rashford converted a late penalty to win 3-1 on the night and progress to the quarter finals on away goals.
United have now won 14 of 17 games since Solskjaer was appointed interim boss, including a 100% record and nine consecutive victories away from home against the likes of Tottenham, Arsenal, Chelsea and now PSG.
The clamour for Solskjaer to get the job permanently is now stronger than ever and Lukaku is clearly one of his biggest fans, seemingly all but confirming the job belongs to Solskjaer.
"I know he is going to stay, there is no doubt about that," he Norwegian broadcaster Viasport.
"He wants to stay, the players want him to stay. We're doing really well and playing like Manchester United should play. He's a young coach, with young players as well, and it's a perfect environment to develop and hopefully win trophies.
"It's always about confidence and belief. When you look at our team, we have a lot of players with a lot of self confidence, but he tells us the right words at the right time.
"That's what you need as a young player, sometimes you need somebody to wake you up, but sometimes someone to give you his arm so you can lean on him, and that's what he does really well. He knows how to find the perfect balance."
The tests keep coming for Solskjaer and United, with a trip to Arsenal scheduled for Sunday in the race to finish in the Premier League top four.