But West Ham fans were left biting their nails at the end as Crewe pulled back a 3-0 deficit.
West Ham forced the pace early on, getting a succession of corners; from one, taken by Matthew Etherington, Marlon Harewood headed just over.
Luke Chadwick had a shot saved before Teddy Sheringham opened the scoring with a first time shot into the top corner after Chadwick deceptively cut back Harewood's pass to him on the quarter hour mark.
He doubled his tally a few minutes later, glancing with aplomb a razor sharp Etherington cross to leave Ben Williams clutching at thin air.
But, as if to serve as a reminder of how Crewe clawed back a four goal deficit at Upton Park last season to grab a couple of goals themselves, Keith Briggs struck a powerful shot from 30 yards that Bywater only just tipped over.
Rufus Brevett, who had earlier whistled a shot well over the bar from distance, tried his luck again and this time, via a deflection, gave West Ham a 3-0 lead.
That was short-lived, however, because two minutes later Dean Ashton received the ball from his captain Kenny Lunt to drill home from close range.
Shortly into the second period, Trent McClenahan, who was on the bench for the first time, came on for his debut to replace Hayden Mullins at right back after the latter pulled up with an injury.
Crewe had a lucky escape when a vicious Marlon Harewood strike rebounded off the post at pace, hit Nigel Reo-Coker four yards in front of goal, and bounced to safety to the dismay of the 1865 travelling fans.
Chadwick was unlucky when he didn't connect cleanly with an Etherington cross and Williams was able to gather from close range.
Harewood was equally devoid of good fortune when he exploited the hesitancy of keeper Williams to push the ball wide.
Crewe then put themselves in with a shout - and lifted their fans - when Dean Ashton found himself one on one with Bywater to poke home.
They nearly equalised with a long punt from substitute Billy Jones which went just wide.
Foster headed a Sorvel cross onto the bar but he was offside anyway - and West Ham survived.
Adam Nowland and Luke Chadwick started the game, while Sergei Rebrov moved to the bench without getting on.
Team: Bywater, Repka, Brevett, Reo-Coker, Ferdinand, Mullins [McClenahan], Sheringham, Nowland, Etherington [Cohen], Chadwick, Harewood.
Subs: Walker, Rebrov, Noble.