Alexander-Arnold put Liverpool on course for their sixth successive league victory this term with a blistering first-half free-kick at Stamford Bridge.
Roberto Firmino doubled their lead before the break, but Chelsea was on top in the second half and got one back through N’Golo Kante.
That set the stage for a tense finish as the visitors held firm under a barrage from Frank Lampard’s outfit to move five points clear of Manchester City and recover from their Champions League loss at Napoli on Tuesday.
“It’s massive for us. We have come here in good form, tried to execute the game-plan and bounced back from a set back in midweek. We’ve shown we are a good side,” Alexander-Arnold told Sky Sports.
“You can’t expect to come here and have it your own way. They are a proud side and were going to come out fighting. The last 25-30 minutes were really tough.”
Alexander-Arnold revealed Liverpool captain Jordan Henderson had considered taking the set-piece before deferring to the England defender, who blasted in for his first goal since November 2018.
“Hendo wanted the free-kick at first and then thought it was too close for him,” he said.
“I thought it was too close to go over the wall, I had to go to the keeper’s side, a good roll and tried to hit it as hard as I can.”
Liverpool had luck on their side as Chelsea’s Cesar Azpilicueta had a first-half equaliser disallowed by VAR for offside against Mason Mount.
Alexander-Arnold admitted it was a significant momentum swing in Liverpool’s favour.
“It is a mental setback for the other team. You think you have scored, celebrate, get back in position and all of a sudden the goal has been cancelled,” he said.