Shaun Miller's last-minute goal for Sheffield United in extra time on Tuesday extended Fulham's winless streak in all competitions to five matches and eliminated the Londoners from the famous knockout competition.
The result also heaped more pressure on Meulensteen with the Dutchman having replaced his compatriot Martin Jol as Fulham's manager on December 1.
But despite having only been in charge of Fulham for a bit over two months, speculation is rife Meulensteen could be on the way out with the Craven Cottage outfit bottom of the English Premier League, although the 49-year-old is adamant owner Shahid Khan supports him.
"That (Khan's support) will only last until that is gone," Meulensteen said after Fulham's loss to Sheffield United.
"So, until that time, I just keep doing my job as best as I can.
"I have no worries in that respect at all."
Fulham last tasted success on January 14 with a 3-0 triumph over Norwich in the FA Cup and Meulensteen has implored his players not to lose faith.
"The most important thing is that we don't lose the togetherness because that's the only way we're going to get out of this mess," the Dutch manager said.
"If you start falling apart within then you have an even bigger problem. That's what cannot happen."
