Despite not being able to add to Ryan Jack’s first-half strike, Davis felt Gers controlled the majority of the 90 minutes at Ibrox.
Speaking to RangersTV after the match, he said: “I thought we had good control in the game but when the score is at 1-0 you always leave yourself a little bit open.
“We had a number of chances and opportunities in the second-half that we could've been a lot more clinical with but overall we are delighted with the three points.
“The goal deserves to be the winner, it was a great strike but we probably didn't test the keeper enough today. We had some good build-up play but that final bit of quality was maybe lacking but I’m delighted and it’s a really hard-earned three points.
Davis, who was voted this afternoon’s Man of the Match, says the whole team worked hard to ensure another clean sheet.
He continued: “I don't remember Greegsy having too much to do and that's testament to the hard work that starts from the front.
“I thought we defended well in the game today overall and as I say the only disappointment is we haven't gone on and tested their keeper a little bit more and made the game a little bit comfortable for ourselves.”
Rangers face Royal Antwerp on Thursday night and Davis is looking forward to being back in European action.
He added: “We are really looking forward to it, Europe has been enjoyable for us throughout my time here and we want to try and go and kick on in that obviously a difficult tie but one we are really looking forward to.”