Summer signing Melligan's misplaced 35th minute cross caught Williams off his line and the keeper could only parry the ball into the roof of the net.
Adam Boyd wrapped up the points from the spot with five minutes to go after Adam Chambers was upended in the box by Gary Roberts.
Eugen Bopp made his full debut for Crewe as Ben Rix dropped to the bench, while Chris McCready's ankle injury allowed Daniel O'Donnell the chance to stake a claim at centre-back.
Orient boss Martin Ling was forced to make one change to the team that beat Walsall last week.
Paul Terry came in for skipper Stephen Purches, who hadn't managed to shake off his hamstring injury.
Crewe thought they were ahead inside the first minute but Chris Dickson was penalised for using Alton Thelwell to climb and head home Billy Jones' corner.
Orient had to wait until the 13th minute for their first sight of goal.
Boyd broke behind the defence but Roberts arrived just in time to head clear his cutback to the unmarked Wayne Corden.
A superb move involving Bopp and Roberts gave Dickson another sight of goal after 17 minutes but the striker dragged his shot wide.
But the O's were ahead on 35 minutes when Crewe keeper Williams was caught napping.
Melligan's misplaced cross caught Williams off his line and the keeper could only parry the ball into the roof of the net.
The O's started the second half brightly with Thelwell steering a header just wide and Corden troubling Williams with a 25-yard effort.
Melligan blew a golden chance to wrap up the points with ten minutes to go but he blasted over from 12 yards.
The midfielder was then the hero at the other end as his last-ditch tackle prevented Tom Pope from bearing down on the O's goal.
But Boyd netted his third goal of the season with ease after Chambers' mazy run was ended by Roberts' late tackle inside the box.