Head coach Meulensteen saw Fulham, who sit bottom of the Premier League, lead for much of the game at Old Trafford through Steve Sidwell's 19th-minute volley.
Late goals from Robin van Persie and Michael Carrick looked to have rewarded United for their dominance of possession and territory.
Yet Fulham hit back in sensational fashion, equalising in the fourth minute of injury time through Darren Bent to secure an unlikely point.
Meulensteen's men host in-form Liverpool on Wednesday, but can now approach that daunting fixture with renewed belief.
"We have got a massive one ahead of us. But hopefully this gives us a good base to kick on," said Meulensteen.
"That's why these players want to play in the Premier League - for these sort of games.
"It's the most exciting league. If you have to come up to the big boys, you have to elevate yourself. You have to step up for it.
"Everybody, no matter where you are in the Premier League, you have got a challenge to play for."
Centre-half Dan Burn shone against United, making a succession of clearances as United peppered Fulham with crosses.
"Since we brought Dan Burn back from his Birmingham loan spell, he's done ever so well," Meulensteen added, before explaining his decision to include the 21-year-old along with fellow youngsters Muamer Tankovic and Ryan Tunnicliffe.
"It's important in the Premier League, if you want to be successful, you need energy and quality.
"That's why we chose some of the younger lads because you know it's a big pitch, big distances to cover.
"Their (United's) game plan was quite straightforward. It was get it wide, get it in. We defended it well.
"If you're well organised and the goalkeeper is in good positions, then it can be easy. You need creativity, variety to open them (opposition teams) up."
