David Moyes' side travelled to Athens having not tasted defeat in European competitions for 17 matches, a streak that included last season's run to Europa Conference League glory, and it is an achievement that no English side has ever bettered.
West Ham could not make it even better themselves, as Konstantinos Fortounis fired the hosts in front and Angelo Ogbonna's own goal just before half-time then doubled the size of the task facing the visitors. Lucas Paqueta's superb volley gave West Ham some hope of rescuing a point with just a few minutes remaining, but that charge came too late from the visitors.
It had been a low-quality opening period in the Europa League clash until Fortounis' opener, which came as he turned away far too easily from three West Ham players in midfield and then toe-poked a finish from outside the box into the bottom corner.
That turned the volume levels up even further in Athens, and the party continued as Ogbonna turned the ball into his own net in first-half stoppage time. The cross was whipped into the middle from the right wing, and at full stretch Ogbonna was only able to slice the ball beyond Alphonse Areola.
Lucas Paqueta, Jarrod Bowen and Michail Antonio were introduced after the break, following Danny Ings blazing over the bar, as Moyes tried to get the visitors back in it, but the sustained pressure did not really materialise for much of the half.
That threatened to change heading into the closing stages, as Paqueta ran onto the ball and laced a volley into the back of the net from the edge of the box, and a further boost soon arrived in the form of eight added minutes. It was not enough though, as Olympiacos just about held firm.
West Ham remain top of Group A, level on points with Freiburg but above the Bundesliga side as a result of their win in Germany earlier this month. Olympiacos in third are just two points behind though, and have the chance to overtake the Hammers when they travel to London in a fortnight.