Clermont led by seven points at half-time, and then the Ospreys lost their Wales fly-half star Dan Biggar through injury, but his replacement Davies took charge, kicking four penalties and a drop-goal following two earier Biggar penalties.
Ospreys moved top of Pool Two as a result of their 21-13 victory, with Clermont not even collecting the consolation of a losing bonus point after centre Wesley Fofana's early try, plus eight points from scrum-half Morgan Parra's boot.
It looked as though it would be a frustrating night for the Ospreys, especially when Biggar limped off after 48 minutes.
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Biggar, his right thigh heavily strapped, tried to run off the injury, but his attempts proved fruitless and the early exit will undoubtedly prove a concern for national coach Warren Gatland three weeks before this season's Six Nations Championship kicks off.
But nothing could take the gloss off a special night for the Ospreys as they extended a remarkable winning record against French opposition in Europe at the Liberty Stadium to 14 games.
Clermont beat the Ospreys in a Champions Cup thriller two months ago, and both teams started where they left off, looking to attack through wide channels.
And such ambition brought an opening try for Clermont after just five minutes, as they went through a number of technically pinpoint phases before fly-half Camille Lopez weaved his way into deep inside the Ospreys 22 and Fofana gathered his scoring pass.
Parra added the conversion, and although Biggar got the Ospreys moving through a ninth-minute penalty, Clermont showed no sign of slowing up, with full-back Scott Spedding proving a particular attacking threat from deep positions.
A second Biggar penalty cut the deficit to a point, but that was immediately cancelled out by a Parra strike after Ospreys wing Eli Walker needlessly infringed through an aerial challenge on his opposite number, England international David Strettle.
The Ospreys enjoyed a sustained spell of pressure after falling 10-6 behind, and their cause was helped when Clermont's Georgia international flanker Viktor Kolelishvili was sin-binned by referee Wayne Barnes after he appeared to thrust his arm in Ospreys skipper Alun-Wyn Jones' face.
Another Parra penalty put the visitors seven points clear, having weathered a storm inside their own half, and Clermont's forwards put together a couple of ominous long-range rolling mauls as half-time approached, suggesting that the Ospreys could face a challenging second 40 minutes.
Parra missed a long-range penalty chance early in the second period, but Clermont were then forced into a change as Strettle limped off to be replaced by Albert Vulivuli.
The Ospreys then lost Biggar, but his replacement Davies then enjoyed a 20-minute purple patch, kicking two penalties and a drop-goal as he inched his team two points in front after Clermont's substitute hooker John Ulugia was yellow-carded.
Ospreys moved to within sight of the Clermont line, and they looked set to claim an interception try when Lopez intercepted and broke clear, but Ulugia had drifted offside and he was sent to the sin-bin for what was adjudged a deliberate offence.
Davies kicked another penalty 10 minutes from time, with Clermont looking a shadow of the side that had dominated during an opening 40 minutes when their forwards were rampant and Lopez controlled matters impressively.
But Davies predictably - and deservedly - had the final say two minutes from time with his fourth penalty that meant the Ospreys scored 15 unanswered second-half points and kept themselves on course for a quarter-final place.
Ospreys pile on the points to stay in European Champions Cup hunt
15/04/2016 AT 21:02
Ospreys pile on the points to stay in European Champions Cup hunt
15/04/2016 AT 21:01