But battling Wolves were rewarded when Cort stooped to steer home a cross from substitute Seol Ki-Hyeon to score his seventh goal of the season.
It was rough on Crewe who looked set to secure their sixth win in seven games after Ashton struck.
Ashton, a January transfer target for Everton, Tottenham, Birmingham City and Aston Villa, took his goal tally for the season to 19 with a first-half strike.
It was another classy display by the in-demand Ashton who was only denied a second goal 11 minutes from time by the woodwork after his powerful drive left the crossbar shaking.
The stalemate leaves Wolves' new boss Glenn Hoddle still searching for his first win as well as leaving them well off the pace in their bid to push themselves into play-off contention.
Ironically, Ashton's goal came after a spell of concerted pressure from Wolves.
Kenny Miller was denied twice - first by a fine save from goalkeeper Clayton Ince and then from a brave block by Neil Sorvel after a cross from Shaun Newton.
Newton turned provider again in the 36th minute with another deep right wing cross.
Miller rose above the Crewe defence only to see his close-range header crash to safety off the post.
It was a let-off that Crewe took full advantage of just two minutes later.
Wolves were caught napping by a quickly taken free-kick from David Vaughan.
Vaughan rolled a pass into the path of Ashton and he beat goalkeeper Michael Oakes with an angled shot from six yards that was helped into the net by Wolves' central defender Joleon Lescott.
It would have been an advantage that was short lived if it had not been for Ince.
Four minutes into the second-half, Mark Kennedy fired a left wing cross into the Crewe area.
Seyi Olofinjana met the centre with a powerful header only to see Ince spectacularly turn the ball away with a one-handed save.
But there was nothing he could do in the first minute of stoppage time when Cort headed home Wolves' face-saving equaliser.