United claimed a dramatic 3-1 win on Wednesday at Parc des Princes to reach the quarter-finals on away goals after a 3-3 aggregate draw.
Trailing 2-0 from the first leg and without 10 senior players due to injury and suspension, United were 2-1 up thanks to Romelu Lukaku's double before Marcus Rashford struck an injury-time penalty in the second half that was awarded following a VAR review.
After playing his part to help United become the first team in Champions League history to progress in a knockout tie after losing the first leg by two or more goals at home, Smalling is confident that the win will propel them through the season's closing stages.
"After Rashford had scored and we had all gone into the corner, I think I was shouting that much that I felt dizzy," he told Manchester United's official website.
"I went into the changing room after and spoke to Luke [Shaw]. I was saying to him how I felt dizzy and he said the same thing as well.
"It was an unbelievable feeling and an unbelievable moment that will galvanise us until the end of the season.
"I think their players thought there was no way, going into the game, that they could lose. But we knew we had a game plan as a team to frustrate them at times and know that we would get chances on the counter attack.
"I think we could have scored a couple more goals as well. Ultimately, they might have had more of the ball, but I felt like we had the better chances. Full credit to the whole XI and the game plan, to stick to it despite times when we came under pressure.
"I think it all just summed up what this club is about. I think it was a great time to be part of this club, for sure.
"Obviously, there was no trophy at the end of the game, but in terms of games and moments in my career, barring cup finals, that is definitely up there."
Ole Gunnar Solskjaer's side travel to Emirates Stadium on Sunday to face Arsenal, who are a point and place behind them in fifth.