Goals from Neill Collins and Michael Doyle gave the Blades a first victory since the opening weekend of the campaign.
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 - the second caution coming following a scything challenge on Ryan Flynn.
Tempers continued to fray as Adams and Jennison Myrie Williams were involved in a furious touchline row when the forward was substituted during the ensuing reshuffle.
"I've told the lads that I didn't like seeing that," Adams said. "I've reminded them that if there is a point to be made then it should be made in the dressing room not out there on the pitch for everyone to see.
"But I was disappointed with the discipline we showed to begin with. We gave away too many free-kicks and the best way to win games is by keeping 11 men on the pitch. We didn't do that.
"We got a goal back and we frustrated them. The 10 men who stayed out there should be proud of themselves.
"But I don't like being called unfortunate because that doesn't sit well with me. We keep losing games by the odd goal and that's not good.
"I honestly believe that if we'd kept 11 players out there on the pitch then we'd have gone on to win the game.
"Why? Because I don't think Sheffield United are a very good side. Whoever comes in has got a very tough job on their hands because their squad of players just isn't good enough. That will sound like sour grapes but it isn't."
- Sports & Recreation
- Jennison Myrie Williams
- Michael Doyle