Glennon saved a first-half spot-kick from Ryan Lowe then pulled off another stop ten minutes from time from Gary Roberts.
Referee David Foster ordered the penalty be retaken and this time centre-back Julien Baudet tried his luck. He too watched helplessly as Glennon smothered his shot.
In the latter stages Glennon did his best to salvage a point for the home side by going forward for a corner.
He stayed up for a subsequent long throw and flicked on with his head but there was to be no fairytale ending.
Crewe, who have now won six of their last eight matches to raise hopes of a late play-off surge, held out to claim maximum points.
Huddersfield went ahead through Chris Brandon midway through the first half before David Mirfin put through his own net on the stroke of half-time.
Lowe made up for his penalty miss when he fired home the winner in the 69th minute.
Huddersfield striker Paul Hayes had plenty of chances in the first half and beat keeper Ben Williams with a drive which came back off the far post.
Brandon opened the scoring with his first of the season in the 22nd minute. He took the ball from Gary Taylor-Fletcher, side-stepped a defender and slotted left-footed in the bottom corner.
Crewe, who had shown little up to that point, came back into the game as Michael Higdon forced Glennon into a save at the second attempt.
A minute from the break, Crewe should have been level when Mirfin cut down Dan Woodards for the first penalty award.
Lowe stepped up to drill the spot-kick down the centre of the goal allowing Glennon to block with his legs.
Crewe were back in the game on the stroke of half-time with a goal which left most people in the ground bemused.
The goal was initially credited to Higdon but replays showed that Luke Varney's cross to the near post missed Higdon and went in off Mirfin's head.
After the break, Huddersfield put the Crewe defence under severe pressure, but the visitors broke decisively in the 69th minute and Lowe fired a superb dipping shot past Glennon.
Ten minutes from time Crewe could have sealed it when Frank Sinclair was adjudged to have tripped Varney just inside the box.
Roberts took responsibility but his kick was brilliantly turned away by Glennon. Referee Mr Foster demanded that the kick be retaken and defender Baudet took his turn but also missed.