The East Anglians roared back early in the second-half with efforts from Richard Naylor and Darren Bent levelling the contest before the hour.
Town boss Joe Royle was left to lament his side's finishing as a host of chances went begging.
Such as the gilt-edged opening Bent missed early on when he was played through on goal by strike partner Shefki Kuqi, only to draw home keeper Clayton Ince then drag his shot into the side netting.
Finnish international Kuqi was a major force for Ipswich in the first 45 minutes and gave Alex centre-half Chris McCready a torrid time. He pounced on the defender's weak header back to his keeper in the 15th minute, but lifted his effort wide of the post.
Crewe were on the back-foot for most of the opening 20 minutes, but they broke to take the lead when David Vaughan slipped the ball through on the left of the Town box and Ashton came in from the tightest of angles to hit a low shot in at the corner.
That came in the 24th minute and Ipswich were reeling when they found themselves two down three minutes later.
That man Ashton again, watched by a posse of Premiership scouts, took Kenny Lunt's pass in his stride and from the left of the box finished across keeper Kelvin Davis.
Kevin Horlock wasted a chance to reduce the arrears after 31 minutes when he lifted a right-footed volley over from 15 yards.
And Tommy Miller did likewise shortly after when he failed to connect cleanly with Jim Magilton's cross, allowing Justin Cochrane to scoop the ball clear from his own six-yard box.
Crewe were brimming with confidence and Ashton must have fancied his chances of a quick hat-trick when he cut in from the left and aimed for the near post. But the effort had much less power than his previous strikes, allowing Davis to make a comfortable save.
The visiting keeper needed more agility to turn Lunt's free-kick over five minutes before the break.
Magilton came close to finding the top left-hand corner with a drive from 20 yards as Ipswich went on the offensive in the second-half.
And they pulled one back moments after Bent had fired straight at Ince from six yards. Kuqi flicked on a corner in the 55th minute and Naylor arrived to thrash a half-volley high into the top corner.
Bent then atoned for his glaring miss by volleying in a spectacular equaliser from the left side of the box after 59 minutes.
Vaughan cleared off the Crewe line as Ipswich mounted a late barrage, but the hosts held on for a useful point.