In a closely-contested game, the duo made the difference as they set each other up for clinically-taken goals at the end of each half that proved too good for their hosts.
New Stamford Bridge boss Carlo Ancelotti had seen his team beat Hull City and Sunderland in their opening two games of the Premier League season, and this was the sort of game they had to win if they are to be genuine title challengers.
But they also possessed the game's outstanding performer in Drogba, whose pace and power ensured Aaron Hughes had a torrid afternoon, and his strike five minutes before half-time, which came when Anelka played him in on goal, always looked likely to be enough to defeat a toothless Fulham.
And any hopes of a Fulham fightback were extinguished 14 minutes before the end as Drogba repaid the favour, allowing Anelka to round Mark Schwarzer and put the game to bed.
Roy Hodgson's Fulham side didn't lack for effort, but they were clearly missing the injured Andrew Johnson, whose partnership with Bobby Zamora had begun to bear fruit, with the pair contributing six goals in Fulham's first four games of the season.
Fulham's seventh-place finish last season was largely built on defensive solidity, but they were torn apart on numerous occasions by Chelsea's intricate passing, with the game's opening goal summing up the visitors' qualities.
Frank Lampard received the ball under pressure before managing to flick it to Anelka, who timed his throughball perfectly for Drogba - who had scored seven goals in his previous nine games against Schwarzer - to run on and fire into the far corner.
The contest had been relatively even until that point, but Fulham were deflated by the goal, allowing Chelsea to go for the jugular at the start of the second half.
And in a game of very few chances that saw neither goalkeeper unduly troubled, a second goal was always going to be enough to kill off the game.
And so it proved, with Drogba repaying the compliment to Anelka by playing a slide-rule pass that saw the Frenchman open his account for the season by rounding Schwarzer and passing the ball into the back of the net.