A second string team, with nine changes from the side that was beaten by Bayer Leverkusen on Saturday, needed some time to adjust but Peru striker Pizarro settled the nerves after he fired them into a 12th minute lead following a fine pass from Robben.
The roles were reversed in the 49th minute when Pizarro, in his second spell at Bayern after joining in the close season, headed on for the Dutchman, who only days ago came back from injury.
"A 4-0 might look easy but it was hard for us," Robben said. "We worked very well together and I am proud of the team."
Keeper Manuel Neuer's replacement Tom Starke did well to deny the visitors in the 24th minute.
But Robben's goal four minutes after the restart put them in complete control and Pizarro, the Bundesliga's all time top foreign scorer, crowned his best performance since his return with a trademark powerful header for Bayern's third goal.
Robben sealed their victory by drilling in from 10 yards in the 88th.
Improving Stuttgart also advanced with a confident 3-0 victory over St Pauli.
Two goals in two minutes from Ibrahima Traore and Vedad Ibisevic put Stuttgart, who have won three of their last four league games after a rocky start, in the driving seat. Tamas Hajnal sealed their win before the break.
Bayer Leverkusen had to work much harder and needed a goal in the first half of extra time by Andre Schuerrle to break the resistance of third division Arminia Bielefeld and win 3-2.
Wolfsburg made it two wins out of two matches since the sacking of coach Felix Magath last week, beating FSV Frankfurt 2-0.
Brazilian Diego, who had fallen out of favour with Magath, was in sparkling form and tapped in from close range for Wolfsburg. Dutchman Bas Dost added another.
Fortuna Duesseldorf scored in extra time to beat Borussia Moenchengladbach, who have only themselves to blame after squandering a bagful of chances, 1-0.
Hannover goalkeeper Ron-Robert Zieler saved two spot kicks to hand his team a 4-3 victory over second tier Dynamo Dresden on penalties.
Holders Borussia Dortmund booked their spot in the next round on Tuesday with a 4-1 win over second tier VfR Aalen while 2011 winners Schalke also advanced after beating Sandhausen.