The Bundesliga leader had a taxing schedule leading into this game, having claimed away wins against title rival Borussia Dortmund in the league and CSKA Moscow in the Champions League in the space of five days.
Robben's first-half double sealed the triumph to extend Bayern's unbeaten league run to 39 games.
The hosts were dominant for long spells and enjoyed 78 percent possession, but rock-bottom Braunschweig had chances on the counter-attack and could have scored on a couple of occasions.
Guardiola was particularly happy with Robben as the Dutchman took his goal tally to 10 in all competitions this term.
"I'm pleased," the coach said after the match. "Robben is very important for us and creates many goals. It was a difficult game because we came from Dortmund and Moscow.
"This game was a whole different ball game."
The victory maintains Bayern's four-point lead at the top of the table from Bayer Leverkusen, with Dortmund a further three points behind.
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- Borussia Dortmund
- Eintracht Braunschweig