Morecambe clinched their first away win of the season with a fine display at Gresty Road.
The Shrimps held the advantage at the break with goals from Jim Bentley and Phil Jevons.
Bentley put Morecambe ahead after just six minutes when he headed a Twiss corner past Steve Phillips at the far post.
And Sammy McIlroy's side made it 2-0 on 37 minutes when Jevons superbly converted Twiss's perfect through ball.
Crewe came back into the game but were twice denied by Barry Roche in first half injury time as the giant stopper saved superbly from Byron Moore and Joel Grant in quick succession.
Roche saved well again at the start of the second half from Calvin Zola as the home side pushed forward and caused problems.
Joel Grant dragged a shot wide before Steve Schumacher gace Crewe hope with a header 13 minutes from time.
Morecambe almost added a third when Michael Twiss hit the post from the halfway line after a poor Phillips clearance but the Shrimps held on for a fine win.
Morecambe boss Sammy McIlroy made three changes to the side that drew 2-2 with Carlisle on Saturday for the Shrimps' first ever visit to Crewe.
Ian Craney, Michael Twiss and Dave Artell are recalled with Stewart Drummond and Danny Adams ruled out and Neil Wainwright on the bench.
Morecambe: Roche; Hunter, Bentley, Artell, Parrish, Wilson; Craney, Stanley, Twiss; Jevons, Mullin. Sus: Davies, Moss, McStay, Duffy, Haining, Wainwright, Curtis.
Crewe: Phillips; Brayford, Mitchel-King, O'Donnell, Westwood; Shelley, Walton, Bailey, rant, Moore, Zola. Subs: Legzdinis, Schumacher, Ada, Donaldson, Murphy, Tootle, Davis.
Ref: G Eltringham