Second-half goals from Danny Ings and Pierre-Emile Hojbjerg secured the three points for Saints.
Hodgson said, "I don't think we played anywhere near enough well in the first 60 minutes of this game to merit the victory. I thought Southampton were better and we did not achieve the level of play we were looking for and capable of, but the words that can temper that statement is that we didn't give up or accept that it wasn't to be our day.
“After the penalty save we were livelier and we asked a lot of questions of their defence and goalkeeper, but they answered them very well and we had to suffer the indignity of the 95th minute goal when we're sending everyone forward to equalise, and they break away and score the goal that kills the game off.
“You could argue that we could have got a draw, but I didn't think we did enough over the course of the 90 minutes to merit that."