The former Chelsea man received a straight red just before half-time in the Anfiel clash with Arsenal which ended 1-1 and manager Roy Hodgson said his prized recruit was left 'devastated' after proceedings.
"He is devastated - his debut at Liverpool, fantastic atmosphere, so desperate to do well," he said.
"He was not playing at the top of his game, but was trying hard to get to that level, when suddenly he gets sent off.
"He was saddened by it all and not least because it put us in a difficult situation."
Hodgson thought the decision to hand Cole a straight red was maybe a touch unlucky.
"Of course we now have the benefit of replays on the TV and those replays suggest to me he was a little bit unlucky," he added.
"I don't think he dived to tackle the player, he dived in front of the ball to try to stop it being played down the line.
"The player got his legs caught up with him and maybe slightly twisted his knee.
"I was pleased to see there was no serious damage as he was able to play the second half."