At that stage, West Brom were still finding their feet but, as they came into the game, the muscular Miller was soon making his presence felt up front.
Phillips did well to keep out the big striker's opportunistic header after he broke clear, then recovered to block Koren's follow-up shot.
But the Rovers keeper will know he should have done better with Bednar's low shot, which span out of his grasp and into the path of Morrison, who finished with aplomb.
The home team kept plugging away and Dean Kiely did well to keep out a snapshot from Andrew Williams before Rickie Lambert sidefooted straight at the goalkeeper from close range.
Miller was still posing a threat at the other end, however, and he used his formidable strength to hold off Joe Jacobson before unleashing a fierce shot that took a slight deflection but beat Phillips through its sheer pace.
That could have finished Rovers but they responded in exactly the right way.
Disley flicked on Andy Campbell's corner and Coles, who had timed his run perfectly, found the net with a crisp finish.
The Pirates went on to enjoy their best spell of the match either side of the break but, for all their pressure, it was still the Baggies who created the best chances.
Miller slipped with the goal gaping after he had rounded Phillips, then fired wildly over the bar after he had ran on to Koren's through-ball.
Phillips was still looking nervous too and he flapped horribly at Bednar's header.
Rovers were still pushing hard themselves and were in with a shout until Hinton's mistake allowed Miller to break away and fire into the bottom corner of the net.
When Phillips came off the bench to meet another Koren pass and steer the ball home, the game was all but over.
But there was still time for Miller to complete his hat-trick with a stylish finish, picking his spot from the edge of the area and casually rolling the ball past Phillips.
Bristol Rovers: Phillips, Lescott, Hinton, Coles, Jacobson, Pipe (Walker 72), Campbell, Disley, Lines (Haldane 55), Lambert, Williams (Igoe 81). Subs Not Used: Mike Green, Anthony.
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