Basel joined Group B winners Real Madrid in the knockout phase ahead of 10-man Liverpool with a 1-1 draw at Anfield.
The Swiss giants dominated the first half and were rewarded in the 25th minute as Fabian Frei lashed a powerful left-footed shot into the bottom corner.
"Going through means me, the boys and my staff will write another fantastic page in the story of Basel. That is the dream," Sousa told reporters post-game.
"Such services give us a lot of confidence for us going forward. It was not easy, Liverpool has a great team.
"I do not think Liverpool played badly. I prefer to look at us and think that we have done many things right."
After substitute Lazar Markovic saw red for thrusting a hand in Behrang Safari's direction, Gerrard almost made the seemingly impossible possible with a tremendous free-kick nine minutes from time.
"It was a very intense game and we had wonderful combinations," Sousa added. "The last few minutes were hard to endure.
"Until the final phase, we were the better team on the court. After that it was difficult to control the game as before.
"But hey, that's Anfield. We played against Liverpool, of course, gave us this great team at the end of a few difficulties.
"Next, we want to rest now, we all deserve. The players celebrate this victory, and that they should."
