The 38-year-old former England international dispatched scored twice inside the opening 22 minutes and the hosts' woeful defence succumbed again when Rufus Brevett hit a long-range shot past keeper Ben Williams.
But the Londoners' hopes of running up a cricket score were quickly dissipated by a goal from Dean Ashton and a succession of poor finishes.
In the end Alan Pardew's side were left hanging on after Ashton claimed his second eight minutes from time to set up a nervous finish.
Early in the game West Ham swept forward with Matthew Etherington exposing Crewe's right flank and Marlon Harewood's burly physique and power terrorising Crewe's central defensive duo of Richard Walker and Steve Foster.
Harewood thumped Etherington's ninth minute corner onto the top of the woodwork, before sauntering through the Crewe back line to set up the opening goal for the visitors in the 16th minute.
Harewood laid the ball off to Luke Chadwick on the right and, when he cut the ball across the box, Sheringham struck a great right footer into the top right-hand corner.
The veteran picked up his second in the 22nd minute with a deft flick of the head to turn Etherington's cross beyond home keeper Williams.
The Railwaymen were camped in their own half for much of the first 45 minutes, but on-loan right-back Keith Briggs did charge up field and force Stephen Bywater to tip over his powerful drive.
The keeper's opposite number, Williams, had no answer though when Brevett came up with a ferocious 30-yard effort on the half-hour mark. The ball flew into the top corner with the aid of a deflection and the Hammers appeared to be out of sight at 3-0 up.
Within a minute Kenny Lunt back-heeled the ball into Ashton's pass and the England Under-21 striker picked out the bottom corner with a low shot.
United spurned several good openings early in the second half with Harewood the guilty party on more then one occasion. However the former Nottingham Forrest man almost broke the woodwork with a thundering volley that crashed off Williams' near post, following some sloppy defending from Walker.
Etherington tucked a one-on-one opportunity the wrong side of the post and Chadwick mis-hit his close-range volley into Williams' hands.
They were almost made to pay when Ashton beat the offside trap and finished a Lunt pass by poking the ball into the bottom corner in the 82nd minute.
In injury time, substitute Steve Jones' overhead kick was far too close for comfort for a side who will expect to pick up maximum points from trips such as this if they are to retain their promotion credentials.