Perez was reunited with his old friend and colleague Andoni Iraola on the sidelines at Anfeld, and it was a perfect start for the LaLiga side, who stormed into a 2-0 lead inside the first 30 minutes on Merseyside.
However, the hosts rallied and brought the game to 2-2 before the interval, and second half goals from Alexis Mac Allister and Florian Wirtz sealed victory for Iraola.
The result continues a mixed start to life for Perez in charge of the Yellow Submarine as he admitted his players missed a key opportunity.
"There's no such thing as a perfect game, and I'll have to be self critical. It's clear that we weren't up to the challenge of a match like this.
"Very simple details, like not generating emotional momentum, which we know is instantly triggered in this stadium by any favourable action towards Liverpool. We couldn't manage it, they equalised, and then at half time, you try to prepare for what's coming.
"They're a high level team. In the end, we managed to stay in the game, but I think we didn't know how to play the key phase, which was right before half time, and from then on it's difficult."