The Greek international gave his side a first-half lead from close range before Aleksandar Mitrovic missed from the penalty spot as Anderlecht spurned a golden opportunity to draw level.
Despite the hosts creating a number of chances, Mitroglou struck again and then put the seal on a fantastic result with a stunning long-range effort that secured the Greek champions their first win in Group C.
The hosts continued their bright start and should have been ahead after 12 minutes when Dennis Praet poked an effort wide of the post before Massimo Bruno smashed a volley into the side netting a just a couple of minutes later.
Anderlecht were soon made to pay for their wastefulness after the quarter-hour mark when Vladimir Weiss cut inside from the right before laying off to Saviola, who threaded a delicate ball into Kostas Mitroglou to finish from close range.
Mitroglou's goal failed to deter the hosts, who nearly responded straight away as Bruno forced Roberto into a smart save from long range.
And they were given the perfect opportunity to draw level after 25 minutes, when Sacha Kljestan was brought down in the box by Andreas Samaris, but Mitrovic’s tame spot-kick was kept out well by Roberto.
Anderlecht continued to probe, with Tielemans’ dipping free-kick saved comfortably before Praet fired over just after the half-hour mark.
Mitrovic should have levelled with two chances in as many minutes just a few moments later before Mitroglou popped up once again to double the visitors' advantage.
The Greek international did well to get on the end of a perfectly-floated ball over the top from Delvin N'Dinga before rifling a half-volley past Thomas Kaminski.
