Alexander-Arnold proved Liverpool's matchwinner at Leicester City on Sunday - as a second-half substitute.
And Carragher insists there's no reason for the situation to change so long as Alexander-Arnold continues to allow his contract to run down.
He told Sky Sports: "You could look at Trent Alexander-Arnold's goal celebration both ways. It might just be that real joy and sensation of scoring such a vital goal or bit of an anger in there for the criticism he has taken because of the contract situation. But he looks like a happy man. It's a great moment for him."
Carragher insists Conor Bradley should keep his place ahead of his senior teammate.
He continued: "If Trent hasn't committed - which he hasn't, obviously, to the manager for next season - he shouldn't be starting games.
"Liverpool are not here to give someone a sendoff... In Conor Bradley, you've got a young player. It's not like we're talking about a 27, 28-year-old, we're talking about a young player, who had a tough 15-20 minutes at the start of the game. And that's part of becoming an experienced player."
Bradley needs his chance
Indeed, Carragher insists Bradley needs the chance to prove himself a worthy replacement for Alexander-Arnold.
He said, "I went through that in certain games, Gary (Neville) who's commentating will've gone through situations like that. It's not about, 'Oh, let's give Trent a sendoff'. It's about, 'Okay, we need to assess whether Conor Bradley is the guy for us next season playing week in, week out'.
"The decision Liverpool will have in terms of Trent moving on is, 'Does Conor Bradley become our first-choice right-back and we buy a backup player who's maybe a utility player who can maybe play centre-back?' Or are buying a first-choice right-back and Conor Bradley's gonna have to battle with him?'.
"So, that lad needs as much experience as he can possibly get. Yes, bring him (Trent) on as you did today if you need to go and win the game, maybe against Spurs or a couple of weeks after that. But in terms of starting games right now, if Trent hasn't committed to the club - which he obviously hasn't - Conor Bradley should be starting every game."