United started slowly but came close to scoring when Che Adams broke down the right, but was denied by Chris Neal in Vale’s goal.
The hosts were growing into the game, and twice had the ball in the back of the net, Billy Sharp the latter to be flagged offside.
Chris Basham broke free, but opted to shoot rather than use several team-mates supporting him, and his shot was deflected wide.
And United were denied by a goal-line clearance when Dean Hammond’s header was hacked clear.
The goal United deserved camed five minutes after half-time.
A low cross from the right found its way to Sharp, who turned smartly before hammering home beyond a stranded Neal.
Despite a few nervous moments, the Blades held on for a home victory - their first of 2016.
Adkins said: “It’s a very good win. We’ve changed our shape going into the game and I felt at times we played some good interlinking football and created chances.
“We’ve scored after building from the back, Billy Sharp’s hit a fantastic strike and it’s a brilliant goal.
“You could see there was a chemistry between Billy Sharp and Che Adams. With the quick interlinking play, you think something’s going to go and happen every time they get the ball.
“We all share the same common cause - we want Sheffield United to do well and today has demonstrated that.
“We’ve endeavoured to play good football, we’ve created chances and the supporters have got right behind the players.
“Today, we’ve seen a united football club and there’s a genuine desire and belief that we can do well. The play-off position is still in our grasp.”