The Millers could not have had a better start with a goal after just 70 seconds. Ben Pringle's free-kick was headed clear by Jay Emmanuel-Thomas but the ball was collected outside the area by Tavernier who side-stepped his marker and found the back of the net with a 25 yard drive.
But despite several good chances, the home side could not build on their lead. Leading scorer Kieran Agard twice fired just wide and also went close with a header from a Nicky Adams cross. Adams also saw a goal-bound shot blocked.
City's only first-half effort on target was a Martin Patterson header after 30 minutes, which was easily saved by Adam Collin.
But three minutes into the second half City were level. Wade Elliott collected what seemed like a harmless ball wide on the right and with Rotherham backing off, he cut inside and hit a fierce shot from just outside the area which beat Collin all ends up.
But Tavernier restored Rotherham's lead on the hour mark. Goalkeeper Frank Fielding was penalised for carrying the ball outside his area and Tavernier whipped home the free-kick.