Thelo Aasgaard's second half goal proved the difference on a day where the Gers created opportunities aplenty but failed to find the clinical touch to create a bigger scoreline margin.
However, it was a priceless triumph for the side who condemned the Lanarkshire side to their first defeat since October 18th.
Speaking to RangersTV after the victory, the Head Coach commented: "I'm very happy for the group.
"We put a lot of effort in and you saw at the end of the game that we were empty. It was a good game from both sides and very enjoyable if you're a neutral supporter, but if you're the coach of Rangers then you have the feeling that you should have killed the game earlier.
"There were a lot of opportunities to make it 2-0 or 3-0. We didn't start well in the first 15 minutes, then we changed a tactical part that helped massively.
"There was a disallowed goal in the second half and then we scored to make it 1-0, but you want to score more because you never know.
"I thought it was exceptional from Jack [Butland] today, especially after the criticism from the Hearts game but I totally believe in him and he helps us to the clean sheet.
"It's now about the next game. It will be a difficult one again. It will be a different challenge from today and it's a team who are very well organised, wait for transition moments, and are very aggressive.
Aasgaard was a bright performer for Rangers throughout the 90 minutes, as was Djeidi Gassama whose effort was parried to the Norwegian international to tap home in the second period.
Röhl was also delighted to welcome both John Souttar and Lyall Cameron back onto the pitch late in the contest with the pair having been sidelined through injury in recent weeks.
He added: "I think Thelo showed good training sessions in the last couple of weeks.
"Djeidi Gassama was really on today, and Lyall [Cameron] deserved to come on because his performance has been right in this training week.
"It's important that everyone plays this role. It's sometimes tough, but today I'm proud of the group.
"I think every player who comes back from injury helps the group. We have had a lot of injuries in the last four or five weeks, and now that we have a busy schedule everyone is helpful.
"It's also good to bring John back. He's a big personality on and off the pitch and he will help us a lot."
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