Taylor’s goal five minutes before half-time was enough to give the cash-strapped Millers and their fans something to cheer about.
Rotherham were the brighter side in the first half and could have taken the lead on 16 minutes but Tom Cahill’s shot on the turn from inside the box was saved by Phil Barnes.
Mark Robins’ side did take the lead through Taylor as he pounced on an under-hit back header from Grimsby defender Ryan Bennett to nip in before the dive of Barnes and squeeze a left-footer in from a tight angle.
The second half brought more of the same as a tired Grimsby failed to cause any problems.
And the Millers could have had a second when Peter Holmes’ 18-yard shot was turned away by Barnes.