Skipper Michael Flynn missed a stoppage-time penalty that would have earned a point for the home side, but they were denied by Logans heroics.
Bradford were heading for a 3-1 defeat when Cheshire referee Mark Heywood awarded Bradford a spot-kick in the last minute after Jason Taylor handled the ball.
Flynn scored from 12 yards, but when Bradford were awarded a second penalty deep into stoppage time after substitute Tom Newey was also penalised for handball, visiting keeper Logan dived to save the Bantams' captain's under-hit effort.
Rotherham went in front after 21 minutes when Bradford left-back Luke O'Brien handled Brett Williams' cross and Lewis Grabban slotted the resulting penalty.
Bradford responded well and deservedly equalised after 34 minutes, with Chris Mitchell heading home Jack Compton's left-wing cross.
However, the match was ultimately settled in a seven-minute spell early in the second half, as Rotherham established a 3-1 lead.
Alex Revell restored the Millers' lead with a low shot just inside the far post after 50 minutes and substitute Marcus Marshall added a third seven minutes later when keeper Jon McLaughlin could only push Danny Schofield's shot into his path - giving the visitors a suitable cushion with which they could withstand the hosts' late onslaught.
Source: DSG