The Robins had striker Billy Paynter sent off in the 74th minute after he led with his arm in an aerial tussle with Michael O'Connor.
But struggling Crewe could not carve out a chance which would have earned them their first win since New Year's Day.
Swindon keeper Petr Brezovan palmed over a late Byron Moore shot and kept out Julien Baudet's downward header with his feet which capped an excellent afternoon for the Slovakian.
The first half was dire entertainment for the 4,000 plus crowd with neither side capable of mounting much in the final third.
Home striker Nicky Maynard was the liveliest performer on display. He nearly nipped in to score in the second minute when he charged down Brezovan's clearance.
And he forced the keeper to save his right-footed effort during a flurry of Crewe pressure towards half-time.
His follow-up effort from that chance triggered penalty appeals and the ball appeared to hit Jamie Vincent's arm but referee Carl Evans ignored them.
Swindon started the second half in the ascendancy and Paynter forced Williams to tip over as well as getting in a header which flew just wide.
Crewe came back in to contention and carved out their best chance of the game when Maynard's harrying forced Hasney Aljofree into a mistake and big striker Tom Pope was left in the clear.
But from a difficult angle, the former Midland Alliance striker guided a shot across the face of goal.
But when Paynter got his marching orders the game looked to have turned in Crewe's favour.
However, despite home pressure Swindon substitute Christian Roberts posed a threat on the break and forced Williams into a near post save.