The Dutch midfielder impressed throughout the opening hour before receiving a second yellow card for fouling Jeremy Doku.
Speaking on Sky Sports, Carragher argued that Kavanagh's earlier decision to caution Gravenberch for a challenge on Elliot Anderson was the bigger mistake.
"It's such a poor decision by the referee," Carragher said.
"The thing that frustrates me is that he wasn't going to give a yellow card. He only gives it when he thinks Anderson is hurt. That has nothing to do with whether a foul should or should not be a yellow card."
The dismissal proved costly as Liverpool subsequently conceded three goals, suffering a heavy defeat at Anfield.
The Reds, now ninth in the standings, will hope to bounce back against LASK in the Champions League, while City prepare to face Paris Saint-Germain in their next European fixture.