Sousa: 'It was glorious'
Marco Streller's scrappy 52nd-minute goal gave the Swiss hosts an important UEFA Champions League win in Group B.
Sousa said the confidence gained from the victory could lift Basel to new heights.
"We have given ourselves a huge amount of confidence with this success. It gives us a boost," he said.
"We were proactive in the game and didn't just wait for them. That was decisive. Our fans gave us tremendous support.
"It was glorious, historic – for me, the players, the fans and the club."
Real Madrid (six points) are clear atop the group, with Liverpool (three, +0 goal difference) and Basel (three, -3) chasing.
Sousa felt his team "could have won by more" against Brendan Rodgers' struggling outfit.
"I'm very happy. It's a great win. I knew we needed a great performance from each individual player if we wanted to beat a team like Liverpool – and we did that," he said.
"We defended well, played well and had chances to get more goals. We could have won by more … against Liverpool. Look at the players they have playing there."