The Bavarian giants advanced two spots to the Bundesliga summit as a late flurry of goals eased them to victory against their 10-man visitors
Franck Ribery opened the scoring on 22 minutes, but substitute Markus Rosenberg drew the visitors level seven minutes into the second half.
Robben, who also began the match on the bench, made it 2-1 from the penalty spot with 21 minutes to go, and Ribery put the game out of reach on 77 minutes as he completed his brace.
Robben rounded off his brace on 83 minutes as he slotted home a second penalty, won by his left-sided counterpart, Ribery.
Early on, the game was played tightly through the midfield, with chances hard to come by. The hosts used the counter-attack to their advantage, though, and had a couple of chances on goal in the first half.
The Frenchman nearly assisted Philipp Lahm with a glorious ball, but Ribery eventually made it 1-0 as he latched onto an inch-perfect ball from David Alaba, took a touch and blasted home off the inside of the left post.
The hosts' lead was never safe at 1-0, though, and Bremen found their way back on to level terms on 52 minutes as Rosenberg latched onto a clever ball from Claudio Pizarro, and blasted a low strike into the lower-left corner.
Bayern finally hit their stride late in the game, and after Andreas Wolf was penalised for a foul on Thomas Muller, Robben slotted home the resulting spot kick, before Ribery gave the home side an unassailable lead.
The hosts pressed on, and Mario Gomez had a seemingly onside goal called back before Ribery was felled by Sokratis in the box, and Robben made it 4-1 as he fired a low strike into the net from the spot.
To make matters worse for Bremen, midfielder Aaron Hunt was sent off for a straight red card on 81 minutes after he committed a dangerous challenge on Toni Kroos.
The result sees Bayern leapfrog Borussia Monchengladbach and Dortmund to move into first place, after the top two sides drew 1-1. Bremen remain fourth, four points behind the Borussias and a single point ahead of Schalke, who have a game in hand.