It followed the Ospreys' 49-19 demolition of Gloucester and Cardiff Blues' 36-20 success against last season's Anglo-Welsh cup winners Wasps.
Newport Gwent Dragons were the only team of four Welsh challengers to come unstuck as they slumped 41-17 to Leicester at Welford Road.
The pool stages comprise only three games per team, so further victories next weekend - Llanelli visit Newcastle, the Ospreys travel to Bath and Cardiff face Saracens away - would put semi-final status in sight.
"The Welsh sides have done well so far," said Davies. "But I am sure there is more to come from the English teams - there are some good sides out there.
"I was pleased with certain aspects of our game and not so pleased with others, but overall we showed a lot of our all-round skills.
"We were sloppy on occasions, which put pressure on our defence, but I thought our players stuck at it superbly in the second half, which augurs well moving forward.
"We know the talent we have got in this squad. We feel we have got a pretty effective game-plan, and we will now look forward to Newcastle next Friday."
Bristol made it five games unbeaten in all competitions by defeating Memorial Stadium visitors Bath 22-15 thanks to a late Marko Stanojevic try.
The Premiership leaders rested most of their league regulars, but still held off a Bath side who must now beat the Ospreys next Saturday to have any chance of repeating last season's semi-final achievement.
"It was an important game for many of those guys as they challenge for a place in our Premiership starting line-up," said Bristol assistant coach Paul Hull.
"They defended really well, so I am sure it will give the selectors a few headaches with their never-say-die attitude.
"The leadership of Matt Salter, as club captain, has filtered right through the squad, and that could be seen especially in the second half as the defence never gave up."