The veteran striker, who netted with both a late equaliser and the deciding penalty versus Bayern Munich in Germany two-and-a-half years ago, obliged from the spot again in a welcome easy win.
Chelsea had begun their European campaign with a lacklustre draw against Schalke and a hard-fought victory at Sporting Lisbon, but they are now firmly on course to top their group for the ninth time in 12 seasons.
The one negative on a comfortable night for Jose Mourinho's side was a groin injury suffered by opening scorer Loic Remy.
With Diego Costa nursing a hamstring problem, the Premier League leaders could now be short of attacking options for Sunday's domestic trip to Manchester United.
Maribor were unbeaten in eight Champions League games this term prior to this, including a win at Celtic and two draws in Group G, but they were outclassed by Chelsea's slick passing and movement on this occasion.
Captain John Terry also netted for the hosts and Eden Hazard scored a late double, while the visitors' misery was compounded by an own goal from Mitja Viler and a missed spot-kick by Agim Ibraimi.
Petr Cech enjoyed his second start of the season, with Thibaut Courtois rested on the bench alongside Gary Cahill, as Kurt Zouma also came in.
Filipe Luis featured at left-back in front Cesar Azpilicueta – sent off against Crystal Palace on Saturday - with Remy up front as Costa and Andre Schurrle missed out through injury and illness.
Chelsea survived an early scare when Damjan Bohar headed over from a Viler cross, but they were in front after 13 minutes when Remy latched onto a throughball from Terry, cut inside and clinically found the bottom-right corner of the net.
However, the France forward suffered a groin injury in making that shot and had to be replaced by Drogba, who doubled the advantage when sending the goalkeeper the wrong way with a 23rd minute spot-kick after Ales Mertelj was somewhat harshly penalised for handball.
It was three on the half-hour mark as Chelsea broke away from defending a corner and a superb move involving Hazard and Cesc Fabregas was rather surprisingly finished off by Terry at the far post.
The fourth came nine minutes into the second period when Filipe caused havoc surging down the left and his cross was cut back by Hazard, before deflecting into the net off Viler for an own goal.
Maribor were offered a chance to hit back when Nemanja Matic bundled over Ibraimi in the box, but the striker then hit the post with his penalty.
England youth international Dominic Solanke was given his Chelsea debut shortly after his 17th birthday when replacing Oscar.
Willian hit the crossbar with a thunderous long-range strike before Branislav Ivanovic went down in the area and this time Hazard coolly slotted home from the spot.
The Belgium international then put the icing on the cake in the 90th minute when racing onto a long throughball from Nathan Ake before firing in his second and Chelsea's sixth.
