Half-time substitute Drogba latched on to a quick free-kick by Frank Lampard in the 73rd minute, hoping to take a touch before firing in a shot.
However, somewhat surprisingly for the Ivory Coast striker, he misjudged the spin on the ball and instead saw it bounce off his right shin and past Jens Lehmann's despairing dive into the net.
The fortunate goal settled a game which neither side deserved to win.
Chelsea defeated Manchester United in their first match of last season and Sir Alex Ferguson admitted it had a dramatic affect on his side's season.
It remains to see if this defeat for Arsenal will have the same effect, but it can't have done their confidence any good and with no worthwhile attempts on goal, Wenger will surly be worried.
It could have been a totally different match had an Asier Del Horno header two minutes into the match finished in the back of the net. The ball flew past Lehman but Freddie Ljungberg was on the line to hack it clear.
Arsenal could have scored within a minute of this let off when, after a fine counter-attack, the ball reached Thierry Henry on the edge of the Chelsea area. However, the normally deadly Frenchman's right-foot curler was poor and failed to trouble Petr Cech.
The striker did better after 12 minutes when he produced a wonderful left wing cross, but Ljungberg, closing in at the far post, half-volleyed it over the bar
Lehmann then knew nothing about stopping a clipped Arjen Robben shot with his face two minutes before half time.
The second half was notable for the substitutions. Chelsea brought on match-winner Drogba at the break, but reserved the last 30 minutes for new signing Michael Essien.
The £24.4million midfielder from Lyon looks to be the final piece in the midfield jigsaw and will, on this performance, be difficult to dislodge.
Also making his Stamford Bridge bow was Shaun Wright-Phillips. The right sided attacker showed plenty of desire and speed, giving Ashley Cole an uncomfortable closing period.
For Arsenal's part, Robin van Persie came on for the injured Ljungberg during the first half, while with the game drifting away with five minutes remaining, Wenger brought on midfielder Mathieu Flamini for Cesc Fabregas.
With Dennis Bergkamp on the bench the move brought into question Wenger's desire to get back into the match.
One plus for Wenger was the performance of new boy Aleksandr Hleb, who looks certain to establish himself as a vital member of Arsenal's midfield.
The visitors had one good chance as they chased the game in the closing stages, but Henry's swerving shot was parried by Cech and it was Chelsea's day.