Goals from Neill Collins and Michael Doyle gave the Blades a first victory since the opening weekend of the campaign and dragged the South Yorkshire side off the foot of the Sky Bet League One table and out of the bottom three ahead of Tuesday night's trip to high-flying Peterborough.
Vale, who levelled through Adams Yates, played the majority of the game with 10-men after Daniel Jones received a red card for two bookable offence after only 20 minutes.
Morgan said: "It's a big thing for us to get that win and a massive thing for us to get out of the relegation zone.
"But what we have to remember is that it is only one win. We have to carry on recording those results and trying to continually improve.
"However, I thought the attitude the lads showed out there was excellent. They really did put in a shift.
"First and foremost, no matter how good you are, that's what you have to do"
Morgan, who is "unsure" when United will appoint a manager on a permanent basis, added: "All I know is that the club are working hard behind the scenes.
"The thing we are focusing on is preparing for and trying to win games.
"Whoever comes in, whoever takes over, we want to leave it in as good a shape as possible. That's the aim and that's what we are working towards.
"No job is easy. Especially when you are at the wrong end of the table as we still are.
"But when you look around this place, see the crowd and the facilities, it's a great job to have and a great club to be at. You really couldn't ask for more."
- Sports & Recreation
- Michael Doyle