The west Londoners languish four points adrift of safety at the bottom of the Barclays Premier League with just 14 matches remaining, including this weekend's trip to Manchester United.
It is a match they head into having reached what Meulensteen admitted was "rock bottom" after seeing his side fall to a 1-0 extra-time defeat to embattled League One side Sheffield United in Tuesday's FA Cup fourth-round replay.
Having fought the tube strike and miserable weather to get there, Fulham fans let their frustrations known at the insipid display - highlighted when Hugo Rodallega was jeered off the field when substituted in the second half.
The Colombia international appeared to take the reception personally as he sat head-in-hands on the bench - an incident Meulensteen downplayed, preferring to focus on keeping together.
"The staff, players and fans are all not happy with this situation as it is," Meulensteen said. "That is normal, that is frustration.
"The most important thing is that we don't lose the togetherness because that is the only way we're going to get out of this mess.
"If you start falling apart, then you have an even bigger problem and that is what cannot happen.
"The most important message to the fans is to not add to the problems we already have, but to be part of the solution and make sure you keep sticking together, keep supporting the team and players most importantly.
"As soon as the players really start to feel that [pressure], then it is an even bigger problem so this is the time when we need the fans the most.
"It is always easy when everything goes hunky-dory and fans are jumping up and down because things are going well.
"No, no, no - that is why fans are called supporters and that is what we need.
"I think they've been absolutely brilliant so far and even today you heard a little bit [of frustration] but it was not over excessive.
"I think everybody can see that we have been a team that has not got the results that we wanted, therefore we have slipped into the position that we're in and we're trying ever so hard to get out of it.
"But there is not a sort of a direct sort of solution and hopefully the new players can help us fix it."
Despite being at "rock bottom", Meulensteen is hopeful the deadline-day additions of players like Kostas Mitroglou, John Heitinga and Lewis Holtby could help turn around Fulham's fortunes.
The bookmakers do not share that optimism, though, making the west Londoners overwhelming favourites for relegation.
"We know it is going to be difficult," Meulensteen said. "We know we have 14 games ahead of us and need to really start to get our act together sooner rather than later.
"Against Sheffield United you were hoping just to breathe some confidence back into the group, with a great opportunity to get into the next round of the FA Cup.
"But it is another blow and another knock you take on the chin. But, with all the optimism I have, we have to keep working hard and we need to get out of it."
While it is all doom and gloom at Craven Cottage, relegation-threatened United head into this weekend's vital clash with Shrewsbury with renewed confidence.
No player needed that boost more than Shaun Miller, whose goal in the dying seconds was his first since suffering a potentially career-threatening knee injury 14 months ago.
Blades manager Nigel Clough hopes the header can act as a turning point for the 26-year-old forward, who was quick to dedicate his goal to his new baby.
"It was an unbelievable feeling to get a goal in the last minute to send us through," Miller told Blades Player.
"The lads were fantastic for not only 90 minutes but the full 120, as well as the previous 90 minutes at Bramall Lane.
"It was a fantastic team performance and to top it off with a goal for my little son, who was born a few weeks ago, was fantastic."
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