Number eight Jake Schatz and flyhalf Quade Cooper added tries for the Reds, who moved to 31 points overall but are still second in the Australian conference behind the Brumbies, who are on 33 points.
Cooper slotted four conversions and a penalty for the visitors while Bundee Aki and Patrick Osborne scored tries for the Chiefs and Gareth Anscombe scored 13 points with the boot.
Both sides were prepared to play the game at a high pace, though the Chiefs were hampered by numerous errors that cost them points and handed at least two tries to the visitors.
Davies, playing in his first game after recovering from a hamstring injury, did not have to wait long for his opportunity when Chiefs flyhalf Aaron Cruden made a break but passed the ball directly to the lurking Reds winger, who ran in untouched to open the scoring in the sixth minute.
The Chiefs got on the board when Anscombe slotted his first penalty, before inside centre Aki finished off a sustained phase of play from the home side.
Anscombe's conversion gave the Chiefs a 10-7 lead after 22 minutes, but the Reds hit back after a perfect rolling maul from an attacking lineout saw Schatz flop over four minutes later.
Cooper converted to give the visitors a 14-10 lead at the break and while Anscombe narrowed the gap to one point early in the second half, the Reds again struck straight back.
The home side were penalised numerous times to allow the Reds to march down field and then hammer away at the line before Cooper sold a dummy and sidestepped his way over.
Davies' second try came from another Chiefs turnover as the Reds quickly counter-attacked with a superb cut-out pass from Will Genia opening up the space down the left wing.
Osborne then provided some impetus for the Chiefs with strong running, which was rewarded with a try before he limped off with a knee injury, but when the home side turned the ball over while hot on the attack with five minutes remaining their momentum was lost.
"We
were getting a lot of opportunities but we didn't deserve to win that
game," Chiefs captain Liam Messam said. "There were opportunities but we
were probably two steps too slow."
Chiefs:
15 Gareth Anscombe, 14 Tim Nanai Williams, 13 Richard Kahui, 12 Bundee Aki, 11 Asaeli Tikoirotuma, 10 Aaron Cruden, 9 August Pulu, 8 Liam Messam (captain), 7 Sam Cane, 6 Tanerau Latimer, 5 Brodie Retallick, 4 Michael Fitzgerald, 3 Ben Tameifuna, 2 Hika Elliot, 1 Toby Smith.
Substitutes: 16 Rhys Marshall, 17 Michael Kainga, 18 Nick Crosswell, 19 Fritz Lee, 20 Brendon Leonard, 21 Andrew Horrell, 22 Patrick Osborne.
Reds:
15 Jono Lance, 14 Dom Shipperley, 13 Anthony Faingaa, 12 Ben Tapuai, 11 Rod Davies, 10 Quade Cooper, 9 Will Genia, 8 Jake Schatz, 7 Liam Gill, 6 Eddie Quirk, 5 James Horwill (captain), 4 Rob Simmons, 3 James Slipper, 2 James Hanson, 1 Greg Holmes.
Substitutes: 16 Albert Anae, 17 Ben Daley, 18 Ed O'Donoghue, 19 Jarrad Butler, 20 Ben Lucas, 21 Chris Feauai-Sautia, 22 Aidan Toua.