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Following their humbling at home to Forest Green in the cup in the week, Rotherham took time to settle against Shrewsbury's long-ball tactics.
They were nearly gifted a goal on 29 minutes when Luke Jones was robbed of possession in his own penalty area by Chris O'Grady who beat Glyn Garner with his shot but it came back off the post.
Stephen Brogan jinked his way in from the left and blazed over and then crossed for Holmes whose header was blocked by Jones.
From a long throw-in Ian Sharps flicked the ball to Mark Hudson whose header was well-saved by Garner.
Then six minutes before the break, Rotherham got the breakthrough when Marc Joseph's cross to the back post was beaten away by Garner.
Pablo Mills headed the ball back towards goal from 18 yards and Taylor got the deftest of flicks to claim his first goal for the club.
Shrewsbury were denied an equaliser when Michael Symes broke down the right and saw his shot blocked on the line by Sharps, and from the rebound Stewart Drummond also had his shot smothered.
Danny Harrison spurned a glorious chance to double the lead when he blazed over the bar with the goal at his mercy at the start of the second half.
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O'Grady hit the woodwork again on 71 minutes with a powerful 20-yard drive.
Rotherham had to wait until the 82nd minute before sealing the points when Holmes headed Joseph's deep far post cross beyond Garner into the corner at a rain swept Millmoor.