Saul struck just four minutes into the first leg of the Champions League last-16 tie as Atletico claimed a 1-0 win to take back to Anfield in the return fixture.
The defending UCL champion had 73 per cent of the possession but failed to manage a shot on target as it suffered only its third defeat of the season in all competitions.
Atleti's performance, inspired by some vociferous home support at the Wanda Metropolitano, was by far its best during a run of just two victories in eight matches since it beat Barcelona in the semi-final of the Supercopa de Espana.
Saul struggled for form in that time and was grateful to the fans for their backing as he delivered a man-of-the-match display.
"It's a result of hard work, of not giving up, of keeping going, even though everyone wants us to down tools," he said. "We keep going on our way, game by game, doing our job and with a great effort by the whole team.
"We did a great job. We were very good defensively. We were united because they have very dangerous players like [Sadio] Mane and [Mohamed] Salah. Being secure on the wings was key to them not progressing.
"There are lots of people that want to kill us and that criticise us, but the true Atletico fans are there in the good and bad [times] and we're grateful, which is the important thing."
Saul admits he was fired up for the match, having been moved back into his preferred midfield role after recently being used at left-back.
"I was really up for it," the Spain international said. "There has been lots of criticism in recent days and I'm happy to have helped the team in my position.
"The stadium was incredible, it reminded me of the old nights at the [Vicente] Calderon."
Atleti faces Villarreal, Espanyol and Sevilla in LaLiga before the second leg at Anfield, which takes place on 11 March.