The right-back completed the full 90 minutes of the Reds' 2-0 Emirates victory, where his 80th-minute free kick was nodded in by Arsenal defender Jakub Kiwior for the game's only goal.
With Joe Gomez covering for the absent Andrew Robertson and Kostas Tsimikas on the left-hand side, Alexander-Arnold was expected to hold his place for Wednesday's EFL Cup semi-final first leg against Fulham at Anfield.
There was no immediate report of an injury to the 25-year-old from Jurgen Klopp after Sunday's game, but assistant manager Pep Lijnders revealed on Tuesday that Alexander-Arnold had joined the Reds' overflowing infirmary.
Speaking to liverpoolfc.com before his press conference, Lijnders confirmed that Alexander-Arnold had hyperextended his knee, and the defender will likely miss the remainder of January as the Reds' rearguard selection crisis worsens.
"First of all, some not-that-good news, Trent hyperextended his knee during the last game. So he has a little tear in the lateral ligament of his knee and he will need time to recover. He had a scan and he will be out for a few weeks, so let's see after that," Lijnders said.
"He will get some rest and then hopefully he can come back to where he was. Because this guy was decisive in all the games and was the one who created constantly for us from deep and gave the team a high level of flexibility. We will really miss him."
As well as sitting out both legs of the EFL Cup semi-final, Alexander-Arnold also stands to miss Premier League fixtures with Bournemouth (January 21) and Chelsea (January 31), as well as an FA Cup fourth-round tie with either Norwich City or Bristol Rovers on January 27.
The England international's participation against Arsenal on February 4 is also in doubt, and in his absence, either Conor Bradley or Luke Chambers could enjoy an extended run in the first team unless Liverpool pursue a new full-back in the January market.
In brighter news, Lijnders confirmed that Virgil van Dijk has made a full recovery from the illness which kept him out of the win over the Gunners, adding: "Virgil is back. He did some runs yesterday on the pitch, looked good again. So he will be back in the squad, he recovered well. So it really helped him not to travel to London."
However, Robertson (shoulder) and Tsimikas (collarbone) are still absent, although Lijnders is hopeful that the pair will be back in team training by the end of the month, saying: "Robbo is almost at the three-month mark, so he will get checked by the surgeon.
"And if he does that scan and we see everything [so] that we can really progress him, he will go to Dubai for a week to do proper volume sessions, proper sessions, so that will be good for him. Then we see the three-month-mark scan and then we can see how quick he can go into contact again.
"Same for Kostas. Kostas is progressing really well, really quick. So hopefully both of them - Robbo and Kostas - will be back at the end of this month to train with the team."
Joel Matip (ACL), Ben Doak (knee), Dominik Szoboszlai (hamstring), Thiago Alcantara (hip) and Stefan Bajcetic (calf) will miss the first leg too, but Lijnders confirmed that the latter two are also on track to resume full training in February.
Liverpool will also welcome the Cottagers to Anfield without the services of Mohamed Salah and Wataru Endo, who are away on Africa Cup of Nations and Asian Cup duty respectively.