Mustafi started the Carabao Cup tie at Liverpool but scored a comical own goal in the sixth minute.
It was one of five goals Arsenal conceded on a thrilling night at Anfield that saw the Gunners crash out in a penalty shoot-out.
Mustafi has made five appearances in the Carabao Cup and Europa League this season but has yet to feature in a Premier League match-day squad.
Former Liverpool defender Carragher told Sky Sports: “I think them days are gone – Mustafi playing in the Premier League.
“He’s just one of those defenders. When someone does that it makes others nervous.
The arrival of David Luiz from Chelsea on deadline day in August further pushed the £35m man further down the pecking order and despite being eager to offload Mustafi, Emery insisted he had full “confidence” in the German.
“I spoke with him when we finished last season,” Emery said ahead of the Premier League opening weekend. “I spoke with him when we started pre-season and I spoke with him also two weeks ago saying maybe there was the possibility he can leave.”
He added: “At the moment he [Mustafi] is here and really if we need, he plays with us. I have confidence he will do well.”