The home side dominated the first half and were only prevented from taking the lead by the woodwork after leading scorer David Mooney struck the crossbar with a header and new signing Charlie MacDonald saw his volley crash against a post before rebounding to safety.
It was no more than Orient deserved when Cook forced them in front in the 26th minute after Crewe keeper Steve Phillips could only palm a Dean Cox cross into the path of the winger.
But against the run of play the visitors equalised after 39 minutes when Moore ran in unmarked to meet a left-wing cross from Luke Murphy and head home from close range.
Crewe improved after the break and both sides had chances to claim all three points with O's keeper Jamie Jones denying Mark Ellis and Moore, while Phillips did well to save efforts from Mooney and MacDonald.
It was only the second time this season the O's had drawn in the league - and both have been recorded against Alex.