The Millers came the closest to breaking the first-half deadlock in the 13th minute when Drewe Broughton chased a long ball, beat his marker and lifted his shot over the advancing Paul Jones. The ball was destined for the back of the net but Troy Archibald-Henville got back to clear on the line.
Exeter had chances of their own, with three chances in as many minutes midway through the first half. First Stuart Fleetwood completely missed his shot eight yards out, before Matt Taylor headed straight at Andy Warrington and then Fleetwood fired over the top from Bertie Cozic’s pass.
Just before the break, Jamie Green’s cross had Jones back peddling but he got a hand to the ball to tip it onto the bar and away for a corner.
But Exeter got the vital goal after 70 minutes when substitute Richard Logan rose at the far post to head home a Rob Edwards cross after a short corner interchange.
And five minutes later Exeter should have made certain when Dale Tonge was sent off for trip on substitute Craig McAllister and City were awarded a penalty, but Fleetwood’s spot kick was brilliantly saved by Warrington.