The hosts looked were comfortable throughout the match, and they took the lead in the 12th minute through new signing Loic Remy.
The Frenchman latched onto a through-ball from John Terry and calmly slotted the ball past the helpless Handanovic.
Jose Mourinho's side doubled their advantage in the 23rd minute when Didier Drogba, who replaced Loic Remy, converted a penalty following a handball from Ales Mertelj.
Blues captain Terry added a third seven minutes later to end the visitors' slim hopes of a comeback.
The home side extended their lead 14 minutes into the second-half after good wing-play from Filipe Luis resulted in an own-goal from Mitja Viler.
Maribor had a great opportunity to reduce the arears in the 63rd minute when they were awarded penalty after a clumsy tackle from Matija Matic.
But Agim Ibraimi spurned the chance to put the vistors on the scoresheet.
Eden Hazard bagged the Blues' fifth from the penalty spot 14 minutes later to rub even more salt into the Slovenian side's wounds.
But their pain didn't end there, as Hazard struck again in the 89th to round off an emphatic victory for the west Londoners.
