All the goals came in a lively first half in a match played between two sides that were separated by two divisions last term.
Alex probably deserved to take all three points as Luke Rodgers and Michael Higdon both had goals disallowed while Ryan Lowe hit the bar.
Veteran Alex boss Dario Gradi handed debuts to summer signings Julien Baudet and Darran Kempson, while Lowe also made his first competitive start for the club.
Alex's Billy Jones saw an early long-range effort deflected wide for a corner before Lowe volleyed a seventh minute effort past Lee Harper. Rodgers almost immediately made it 2-0 but fired narrowly wide.
However, the Cobblers equalised in the 13th minute when Pedj Bojic's low cross was guided home be captain Scott McGleish.
Crewe went ahead again on 21 minutes when Mike Higdon won a flick-on to release Lowe, who whipped in a cross for David Vaughan to power a header home.
The impressive Lowe crossed for Higdon to head narrowly over, before Chris Doig made a goal-saving tackle to deny Rodgers.
Town grabbed a second equaliser four minutes before the interval. Sam Aiston, a summer signing from Tranmere Rovers, was half-stopped by Jon Otsemobor only for the clearance to fall invitingly to the feet of Andy Kirk, who drove home from close range.
The second half was quiet compared with the excitement of the opening 45 minutes although the lively McGleish forced Ben Williams into a smart save.
Kempson narrowly failed to get his head to another Lowe cross while Higdon and Rodgers both went close.
Higdon flicked a header from Lowe's cross just wide of Harper's left-hand post as Crewe pressed for a late winner and Vaughan had late shouts for a penalty after he appeared to be impeded by Jason Crowe waved away.
But both sides eventually had to settle for a point after a bright opening game of the season played in humid conditions.