Liverpool reopen their five-point cushion at the top of the standings, while Chelsea are 11th after seeing their four-match unbeaten streak on the domestic front come to an end.
While both teams had their moments during a lively opening which involved little goalmouth action, it was Liverpool who took the lead in fine fashion through Alexander-Arnold.
Chelsea were forced into an early change as Emerson Palmieri, on his comeback from a thigh problem, needed to be withdrawn, but it did not hinder the Blues who were arguably having the better of the game from open play.
Moments after Tammy Abraham had fired straight at Adrian from a one-on-one opportunity, the home side had the ball in the net through Azpilicueta, who bundled the ball home after good work from Willian down the left.
However, not for the first time this weekend, VAR intervened. Mason Mount's left leg was offside in the build-up to the goal, leading to the equaliser being chalked off.
To compound Chelsea's frustration, Liverpool went up the other end to double their advantage, again from a set piece. Andrew Robertson's cross from the left found Firmino, who powered a free header into the top corner from 10 yards.
Frank Lampard was soon forced into a second alteration at the back after Andreas Christensen picked up an injury while blocking a Salah shot, leaving Chelsea with just one substitution after the restart.
Liverpool went straight onto the front foot in the second half, and it took a stunning one-handed stop from Arrizabalaga to prevent Firmino's volley from finding the far bottom corner.
The Merseyside giants were doing an effective job in keeping their hosts penned back into their own half, although Chelsea soon began to have some more joy with Kante dragging a low effort just wide from 20 yards.
While that acted as the catalyst for Chelsea to find a second wind, they were struggling to create anything of note in the final third as the game progressed into the final quarter.
However, out of nothing, Kante halved the deficit for his team. There appeared little on for the Frenchman when he collected the ball around 30 yards from goal, but the midfielder ghosted into the area before finding the top corner of Adrian's net.
Despite Liverpool remaining a threat once they got on the ball, there was a sense of panic within the ranks of the Merseyside outfit, and it led to an effort being made to break up Chelsea's rhythm.
Lampard's team rallied yet again as the game drew closer to the full-time whistle and after Michy Batshuayi had glanced a header wide of the far post, Mount sent a first-time effort over the crossbar from just 12 yards out.
Liverpool were able to come through four minutes of added-on time with minimal fuss, however, as they put clear daylight between themselves and defending champions Manchester City.
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Christensen (Zouma 42'), Tomori, Palmieri (Alonso 15'); Kante, Jorginho, Kovacic; Willian, Mount, Abraham (Batshuayi 77')
LIVERPOOL (4-3-3): Adrian; Alexander-Arnold, Matip, Van Dijk, Robertson; Wijnaldum, Fabinho, Henderson (Lallana 84'); Salah (Gomez 92'), Firmino, Mane (Milner 71')