Aldershot responded on four minutes, when Grant's thunderous shot from the edge of the penalty box produced a spectacular save from Andy Warrington.
On the 15-minute mark Shots captain Rhys Day stopped a certain goal with a last-ditch tackle as Drewe Broughton attempted to pull the trigger from six yards.
Warrington was then called into action making another good save low to his right from a Kirk Hudson effort.
Seconds later Warrington again denied Aldershot from distance with a fine save, from a rare Anthony Charles strike. On 22 minutes, Broughton again made space, but directed his header over the bar from Omar Garcia.
Three minutes later Charles rocked the crossbar with a sumptuous drive from 30 yards.
Like the first half Rotherham started brightly after the break, Mills picked out Alex Rhodes from the right, but the midfielder headed wide from six yards when it seemed easier to score.
Aldershot did not heed the warning and fell behind to a well worked corner routine.
Rhodes' corner picked out Micky Cummins on the edge of the area and his half volley flew through a sea of players for a well earned lead.
Just past the hour mark Warrington again brought gasps from the crowd with another great save. Initially unsighted he scrambled across the face of his goal to tip Scott Davies' piledriver round the post.
With Aldershot looking increasingly desperate Nick Fenton brought down Marvin Morgan in stoppage time.
From the spot Grant made no mistake to preserve Aldershot's year-long unbeaten home record.