It was a mistake by the experienced keeper that prevented his side beating York at the first attempt in the first round, although they eventually won the replay.
This time it was Mildenhall's agility and bravery that kept Rovers in the competition as Crawley looked to make an instant impression on Gregory.
Ward said: "Steve held his hand up over the error against York and apologised to the lads, which is typical of him.
"If anything he has raised his game again since then. He is a terrific professional and his first words to me at the end of this game concerned it being his third successive clean sheet.
"He has been really commanding and might easily have helped us win the game. We have finished it strongly after a tentative start.
"I wouldn't say we did enough to earn a victory. But we will have John-Joe O'Toole and Chris Beardsley back from suspension for the replay and are not out of the tie by any means."
Gregory's side looked the more likely scorers in the first half and Mildenhall had to be at his best to beat away a fierce volley from Mark Connolly after 43 minutes.
The tall goalkeeper had also done well to save a 15th-minute header from Gary Alexander and in the second half made further saves to deny Dannie Bulman and Joe Walsh.
Oliver Norburn hit the crossbar with Rovers' best effort, a right-footed volley from 20 yards just before the hour-mark.
Crawley goalkeeper Paul Jones later had to five full length to keep out a long-range effort from Eliot Richards and the League Two side battled to the last.
The visitors lost skipper Josh Simpson to a hamstring injury in the warm-up and had to draft in Andy Drury as a late replacement.