The Gers made a winning start to their campaign thanks to a goal in each half by new boy Graham Dorrans, with Ben Heneghan on target for the hosts five minutes before the break.
At 2-1, there was always chance that Well might have forced an equalizer, but Pedro Caixinha's side held on to claim all three points.
The Glasgow giants were dominant in the first half but failed to capitalize on having the majority of possession, and that is an area in which they must improve, according to the veteran hitman.
"The final third is always where games are won and lost and if we had made better decisions in that 20, 25 minutes, there were three or four occasions where if we had made that right pass and been a bit more clinical then we could have been two or three up," said Miller.
"That is what happens and you make it hard for yourself. It was a hard fought win, it is always tough to come here. I think there are definitely positives from that first half, some of the play was great and we should have been a couple up."