Rovers, taking part in their first quarter-final for half-a-century, were in the tie for a long spell before, finally, being swept aside in the last 20 minutes.
West Brom, after a round of cup shocks, were concerned that they could have become another victim of a lower league coup.
Early on they were indebted to the positive defending of centre-back Leon Barnett, but finished the game well on top.
From one of Barnett's many tackles, Rovers had looked hopefully for a penalty when Andy Williams went down but, after a long look, referee Mark Clattenburg waved the claims aside.
When West Brom hit their stride their speed swept them into a two-goal lead on the half-hour.
It should have been all over then but, within two minutes, their sloppy approach to Rovers' first corner saw the home side grab a goal back through Danny Coles.
The first goal had stemmed from a moment's hesitancy in Rovers' midfield by Craig Hinton which saw Roman Bednar race past him to thump a shot at Steve Phillips. The keeper scrambled that shot away one-handed but only as far as James Morrison.
The second goal was even more demoralising for the League One side as Miller raced past young Welsh full-back Joe Jacobson to drive a shot beyond Phillips.
Rickie Lambert had earlier passed up a simple chance to score his season's sixth cup goal but, after Coles had scored, Rovers had a good spell in the second half.
The tie turned cruelly against them when Miller was played onside for his second goal. Former England international Kevin Phillips came on to score the fourth and Miller coolly then completed his hat-trick with five minutes to go as the Pirates fine run ended in defeat.