Emmanuel Fernandez’s towering early header put the Light Blues ahead before Tete Yengi equalised for the visitors.
But Mohammed Diomande’s goal with 12 minutes remaining gave Röhl’s men all three points to make it four straight Premiership victories.
The Gers gaffer knows his side weren’t at their best but insists picking up another win was the most important thing.
Speaking to RangersTV, he said: “I am very happy with the three points, I think it was a hard-fought victory and it was not easy.
“Before we went 1-0 up, we were good in the game, we played fast, exciting football, and then we slowed down, and we weren’t clinical with some of our decision-making.
“At 1-1 we lifted our opponent, and then it is tough to play against a deep block, and you have to play fast and attack the last line more.
“We changed things at half-time and I was happy with the impact players as they gave us a lift, which is good.
“I told my players we need everyone, and they will all get a chance.
“All my players are ambitious, and they know if it is a good game or not. It is my job to show them solutions, but I want everyone to put everything in.
“I said in my pre-match press conference you need dirty wins sometimes and this is what it was.
“We want to do better, but we will take the win.”
Diomande’s strike was initially ruled out for offside before a VAR check overturned the decision to see the goal awarded.
Röhl was confident the Ivory Coast midfielder was onside to keep up his perfect record in the league.
He added: “We attacked the last line and made the run. I saw the full-back was a bit behind and I had a good feeling it wasn’t offside.
“We hoped Livingston’s goal was offside, but we made our goal and we defended well at 2-1.
“We had a few moments to make it 3-1 and then it becomes a different game as they have to open up and we get more space.
“But we will take the win and we know what we need to do to improve.”
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