Aaron Callaghan voiced his frustration as an own-goal condemned Carrick to another morale-sapping defeat.
Sam O’Connor’s error with 20 minutes left settled an otherwise forgettable game.
It was Rangers’ 18th defeat in 26 league games this season, keeping them deep in relegation trouble.
Callaghan felt the loss was harsh on his side.
“I don’t think we deserved to get beaten today, I don’t think any team deserved to win, and I think any neutral will tell you that,” the Carrick boss said.
“It had a draw written all over it - very few chances, I don’t recall any outstanding saves from the goalkeepers.
“I felt there was nothing in the game - 0-0 at half-time was a fair scoreline, and it probably should have finished that way.”
The only goal came in the 70th minute.
Ally Teggart’s cross to the back post caused panic in the Carrick back line, with O’Connor directing the ball into his own net.
Callaghan added: “It’s very frustrating because you prepare all week on a game plan, and I thought it was working.
“We had the outstanding player on the pitch today in Eamon McAllister. I thought he was absolutely brilliant.
“To concede in the way that we did, where Sam is running back towards goal and it hits off his knee or his shin - that is the rub of the green that goes against you when you’re down at the bottom.”
A miserable day for Carrick got worse when they were reduced to 10 men in the 77th minute.
Aaron Smyth, already booked in the first half, went in strongly on Jarlath O’Rourke, drawing a second yellow.
Callaghan added: “I don’t think he needed to send the player off. I feel sometimes they give decisions too quickly.
“It stopped us chasing the game. Suddenly we were down a player and down a goal.”
In the away dugout, Rodney McAree felt it was one of Dungannon’s most satisfying victories this season.
“It’s one of our best wins of the season,” he said.
“The conditions were tricky and it didn’t allow for pretty football, but I thought it was one of our best performances in terms of getting the job done.
“I am pleased to get the three points on the board and the guys did really well out.
“We want to keep on improving and getting better and the guys put in a great shift out there.”