Hodgson felt his side deserved their victory, even if there was a slight suspicion that Cole's winner was offside.
"For 90 minutes I thought we were clearly the better team. We had to play well to give that performance and I am naive enough believer that if you give the performances then often you do win," he said.
"I am extra satisfied we got the winning goal but even at 1-1 I wouldn't have been in any way dejected because I think today we put Wednesday's game well and truly behind us.
"Even if you win it to a controversial goal I don't think we should be in any way apologising for it and anyone who begrudges us that victory after watching the game is a very churlish person."