Having already missed a hatful of chances, Anelka diverted Jose Bosingwa's long-range effort past the impressive Paul Robinson six minutes before half-time, before taking his top-flight goal tally into double figures for the season with a deft finish after the break.
Rovers boss Paul Ince made three changes to the side that drew 2-2 at West Brom last weekend, with Zurab Khizanishvili a surprise inclusion at the expense of Christopher Samba.
Morten Gamst Pedersen came in for the injured Brett Emerton, while Carlos Villanueva replaced the suspended Benni McCarthy.
Chelsea made just the one change following their 3-1 defeat at AS Roma in midweek, with Salomon Kalou returning to the starting line-up in place of Joe Cole, who remained in London along with Didier Drogba.
In torrential rain, Rovers made a bright start straight from the kick-off, with Keith Andrews firing high and wide after just 25 seconds.
The Irishman, who scored a last-minute equaliser in the game at The Hawthorns eight days ago, almost went from hero to zero moments later, when his back-pass to Paul Robinson was under-hit.
Anelka intercepted the ball, rounded Robinson and went down under the keeper's sliding challenge, but the strong appeals for a penalty from the Chelsea supporters behind the goal were waved away.
Anelka's flick forced Robinson to tip the ball over the bar a minute later, before the Rovers shot-stopper denied the Frenchman's downward header with his feet. Robinson then made an almost identical save from John Obi Mikel shortly afterwards.
Florent Malouda was next to try his luck, firing well wide on 19 minutes, before seeing his shot deflect narrowly over seconds later.
Frank Lampard's long-range strike was then palmed wide as Chelsea continued to dominate, before Rovers almost snatched an unlikely lead.
A brilliant break by the home side saw Andrews pick out Stephen Warnock, who centred for Villanueva and the Chilean's curler forced a stunning save from Petr Cech.
Chelsea launched a quick counter-attack of their own just past the half-hour mark, with Kalou playing in Anelka, but the former Bolton forward's left-foot strike was sensationally saved by Robinson.
The Premier League's leading scorer did eventually break the deadlock, although he didn't know much about it. On 40 minutes, Bosingwa's speculative strike deflected off Anelka's knee and wrong-footed Robinson in the Rovers goal.
Bosingwa almost gave away a penalty as a frenetic first half drew to a close, but his challenge on Pedersen went unpunished.
Chelsea have put so many teams to the sword already this season, but Rovers refused to simply roll over and the home side had an excellent opportunity to equalise nine minutes after the break.
Pedersen clipped the ball up to Jason Roberts, who spun past John Terry superbly, but his snap-shot was well blocked by Cech.
And the miss proved costly, as Chelsea doubled their advantage with their next attack. Malouda broke from the left, Lampard slid the ball through and Anelka lifted a left-foot shot over the advancing Robinson for his tenth goal of the season.
The second goal proved decisive, as the visitors saw out the remainder of the game with ease. As the clock counted down, Lampard rattled Robinson's crossbar with a fine free-kick, before Anelka was denied his hat-trick by another smart stop by the former Leeds and Tottenham keeper.