The Hammers became only the second side to win at Chelsea in the league this season and Frank Lampard's side, which lost at Manchester City a week prior, could have few complaints about the result.
With Tammy Abraham sidelined and N'Golo Kante among the substitutes, Chelsea lacked a cutting edge in attack and guile in midfield as West Ham's Mark Noble and Declan Rice were given the run of the park.
Cresswell scored the winner early in the second half and West Ham could have extended its lead before surviving a late rally from the hosts to secure its first victory since beating Manchester United on 23 September (AEDT) and provide a lift to under-fire manager Manuel Pellegrini.
Mason Mount drove Chelsea forward, surging through midfield with the ball at his feet, and he almost opened the scoring from the edge of the box but slipped as he struck a left-footed drive into the body of David Martin.
Christian Pulisic and Kurt Zouma both fired wide from close range but Chelsea's defensive frailties were in evidence when Robert Snodgrass was given time and space to whip in a cross that Michail Antonio headed into Kepa Arrizabalaga's hands with the goal at his mercy.
Olivier Giroud had the chance to open the scoring three minutes from half-time when Martin parried Mateo Kovacic's shot into his path but the France international's attempt to flick it over the goalkeeper from point-blank range failed miserably, and Pulisic missed blazed over the crossbar just before the interval.
Three minutes into the second half West Ham exploited the ample space afforded to it in Chelsea's half when Pablo Fornals fed Cresswell, who skilfully cut inside Reece James and curled a shot with his weaker right foot into the far corner of the net with Kepa at full stretch.
Kepa had to be alert to save Fabian Balbuena's powerful header just before the hour mark and Antonio had the ball in the net 10 minutes later only for his bundled effort to be ruled out for handball.
Chelsea only began to show the kind of urgency the situation required when Lampard sent on Callum Hudson-Odoi, whose nimble feet on the left wing caused havoc but failed to create a clear chance as the visitors clung on.