Shaun Miller grabbed the winner in the final minute of extra time, but the Blades did a good job of containing their Premier League opponents throughout the match.
The Cottagers are bottom of the league, four points from safety, and Meulensteen admits that his players have hit "rock bottom".
"We know that we've only got ourselves to blame. At the end of the day, that's what we need to do. We need to look each other in the eye and say, 'Let's pick ourselves up and dig ourselves out of it," Meulensteen told Sky Sports News.
"If you talk about this game and you say, 'Have we hit rock bottom?' We probably have. OK, fine, that's it.
"And if we have hit rock bottom, it can't get any worse and it can only get better, so that's what we need to work for."
Fulham return to league action against Manchester United on Sunday afternoon.