Firmino, Sadio Mane, Mohamed Salah and substitute Daniel Sturridge all got on the scoresheet on a harrowing afternoon for the visitors, who failed to register a single shot on target.
The result means Arsenal only have three points from their opening three fixtures, leaving them languishing in 16th place on the Premier League table heading into the international break.
Carragher said the Gunners board had "bottled" the decision to part ways with manager Arsene Wenger this summer after a season in which the club finished outside the top four for the first time under the Frenchman.
"It's the last 10 years, and you will see the same performances for the next 10 years because there is a lack of ruthlessness from the top to the bottom," the former England centre-back told Sky Sports.
"If we are being honest, the board bottled the decision in the summer and were terrified of moving the manager on. There had to be a change. Then the summer comes and it's 'things may be different'. But what we saw today was worse.
"I called them cowards last season and got a bit of criticism but I'll say it again: cowards. That was worse than what I saw at Crystal Palace (3-0 defeat) last year.
"I love Arsene Wenger but the club were not ruthless enough. Everything is nice at Arsenal, there is no ruthlessness. What have these players done? They should be challenging."
Wenger signed a new two-year deal at Arsenal in May despite coming under pressure from some sections of the club's supporters to resign. He has managed the Gunners since 1996, leading them to three league titles.
The Gunners will return to action from the international break with a home game against Bournemouth on 9 September, before hosting Cologne in the Europa League the following midweek.