The visitors dominated the first half and might easily have taken the lead early on with an Alan Sheehan low drive from 25 yards testing Phil Smith in the Swindon goal.
It was the Swindon keeper again the spotlight when Gilfi Sigurdsson brought a flying save from another well-struck effort.
The visitors might well have taken a lead on 32 minutes when Sheehan set-up a glorious chance for Danny Woodards but he side-footed wide.
In the next attack, Clayton Donaldson saw his glancing header from a Scott McManus cross fly just wide of the far post.
Crewe manager Gudjon Thordarson came expecting a battle.
He said: "We expected a battle but I thought we did well in the first-half. But I was concerned at half-time because we did not take our chances.
"Swindon were stronger in the second half and our keeper did well. I believe three teams are already gone and the fight is on for the remaining safe place.
"It will go all the way and 50 points may not be enough, but we must remain cool and calm.''
In the second half it was Swindon who dominated possession but they were unable to prise open a strong Crewe rearguard.
Top scorer Simon Cox should have done better but he reacted slowly as a cross fell to him in the six-yard box but goalkeeper John Ruddy was able to smother his effort.
Swindon manager Danny Wilson was not too disappointed, saying: "It was disappointing but it's not the end of the world and we have another point.
"We kept a clean sheet and dealt with free-kicks and long throws and were at the top of our defensive game."