Jurgen Klopp's side have seized just one point from their last three outings in the Bundesliga and went into Wednesday's fixture on the back of a painful defeat to arch-rivals Schalke.
However, Ciro Immoblie fired them into a third-minute lead following a handsome throughball from Shinji Kagawa.
A brace from substitute Adrian Ramos then put the victory beyond doubt, a predator's goal preceding a fine individual effort.
Immobile's strike represented his second in as many Champions League appearances for Dortmund, and the club's second-fastest goal in the competition after Andreas Moller's effort against Juventus in 1995.
Dortmund have a maximum haul of six points in Group D, with Arsenal second on three points ahead of Anderlecht and Galatasaray on one apiece.
Belgian champions Anderlecht, who are now winless in 10 Champions League games, welcomed back goalkeeper Silvio Proto from suspension and influential playmaker Dennis Praet following a calf strain, while Sebastian Kehl's return eased Dortmund's injury problems.
Klopp nevertheless elected to drop Mats Hummels, who is short of match fitness, and start Immobile instead of Ramos at his spearhead.
It took the Italy international little time to vindicate that decision. Kagawa's masterful clipped pass sent Immobile through on goal, and the latter stabbed the ball into the bottom corner with customary composure.
Anderlecht's Aleksandar Mitrovic then had a goal ruled out for offside, before Aubameyang immediately spurned a gilt-edged chance to double Dortmund's lead. Kagawa employed exquisite footwork to free Kevin Grosskreutz down the left and, with goalkeeper Proto drawn to him, the Germany international cut the ball across the box for Pierre-Emerick Aubameyang, who somehow fired wide despite half of the net being unguarded.
Steven Defour's piledriver forced Roman Weidenfeller into a smart save as Anderlecht began to get into the game.
Grosskreutz bravely poked the ball into the net after 26 minutes, under the challenge of Proto, but his supplier Aubameyang had drifted offside and the goal was disallowed.
Proto kept out a Marcel Schmelzer effort thereafter, and Anderlecht will have been relieved to reach half-time with only a one-goal deficit.
The second half continued as the first had ended, with Dortmund taking the game to their hosts and Proto being called upon to save a fierce Immobile drive and an Aubameyang attempt.
From a rare Anderlecht attack, Chancel Mbemba did produce a fine shot that flew inches over the bar, but that merely offered brief respite from relentless Dortmund pressure.
Ramos came off the bench in the 65th minute and soon put the result beyond doubt.
Lukasz Piszczek curled a magnificent, low cross around the back of the Anderlecht defence in the 69th minute and the Colombian got ahead of his man to poke it home.
Ramos completed the win by spinning his marker and producing another fine finish with 11 minutes to go.
