The Northern Ireland captain was instrumental in the middle of the park for Gers as despite falling to a Leon Balogun red-card and equaliser before half-time, they found second half goals from Scott Wright and a James Tavernier penalty to take the win back up the M74.
Speaking to RangersTV afterwards, Davis said: "We are delighted. It maybe wasn't how we wanted the game to look, but once we went down to 10-men, it was important to show that character and I think in the second half we looked a threat going forward and solid defensively and we are delighted to take the three points.
"It was important we reset at half-time. The manager tweaked a couple of things and we started the second half well and that gave us a good foothold in the game.
"From then on, we felt relatively comfortable in the game and it was important to keep our shape, limit them and we always carry that threat."
Davis also cast an eye forward to Thursday, and the UEFA Europa League semi-final first-leg with RB Leipzig in Germany.
It is 14 years since Gers have featured in European competition at this stage, with Davis having been involved for the club last time around.
He added: "It is going to be a really tough tie over the two legs but we are in the semi-finals and really looking forward to it.
"Hopefully, we can go that one step further."
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