Lambert's blushes looked to have been spared midway through the second half on Saturday when substitute Nicklas Helenius bundled home to cancel out Jamie Murphy’s early opener.
But the Premier League outfit were sent crashing out when Ryan Flynn powered his way into the area before lashing home for the League One strugglers 10 minutes before the end.
And Lambert admits he was frustrated by his side's performance.
He told BBC Sport: "I don't think we did enough to go through and that is me being honest. Good luck to Sheffield United in the next round. We never did enough to go through.
"We had a lot of the ball and huffed and puffed. It was one of those kind of games.
"I know it (an upset) can happen and it is not a great feeling when things like that happen to you."
Much had been made of the Scot's pre-match comments regarding the importance of the FA Cup, with Lambert denying a lack of respect for the competition.
He added: "I was asked a question which the journalist in question actually regrets asking. He knew the context in which I was talking.
"I never decried the FA Cup in any shape or form. I never demeaned the competition. I respect its history. I put out the strongest team I could do to try to win the game.
"The FA Cup is a brilliant competition and everybody wants to do well in it.
"If you look at my own career, I won things in cup competitions. I won a few leagues here and there but actually won competitions so I am not degrading it."
