The defender was returning to Anfield for the first time since he left in the summer
Alexander-Arnold, 27, walked away from his boyhood club in the summer after turning down the chance to sign a new deal.
He joined Spanish giants Real but made the return to the north west on Tuesday evening after the two sides drew each other in the Champions League.
While some home supporters stood up to applaud him as he came on as a second half substitue, his every touch was widely booed.
The hostile reception followed the defacing of a mural of Alexander-Arnold on Merseyside, which called the Scouse defender a "rat" in Spanish.
The hosts secured all three points thanks to the only goal of the game scored by Alexis Mac Allister.
Speaking before the game, England international Alexander-Arnold admitted he was unsure how he would be recieved by the Kop crowd.
"Whatever way I'm received, is the decision of the fans," he told Prime Video Sport.
"I'll always love the club, I'll always be a fan of the club.
"I'll always be thankful for the opportunities and the things we achieved together, they'll live with me forever.
"No matter what, my feeling won't change towards Liverpool. I've got memories there that will last me a lifetime, and no matter how I'm received, that won't change.
He added: "When the draw was announced, everyone knew it was going to happen, it was destined for that fixture to come."
On Monday, Alexander-Arnold paid tribute to former Liverpool teammate Diego Jota by laying flowers outside Anfield alongside his manager, and former Red, Xabi Alonso.
