Bayern, chasing a sixth successive title, opened up a 20-point lead over Schalke 04 with nine games to go.
Peruvian forward Pizarro became the fourth oldest player to net in the Bundesliga when he gave Cologne an early lead at home to VfB Stuttgart before Germany's Gomez replied with a brace to lead the visitors to a 3-2 win.
Pizarro, 39, put bottom-of-the-table Cologne ahead after seven minutes and has now scored at least one Bundesliga goal in each of the last 20 years.
Cologne dominated, threatened a second but were floored when Gomez, 32, scored twice in three minutes before halftime.
Andreas Beck added a third for Stuttgart in the second half before a Milos Jojic free kick gave the Billy Goats a glimmer of hope five minutes from time but it was not enough.
Pizarro has never taken part at a World Cup as Peru have not qualified since 1982 and he last played for his country in March 2016, but coach Ricardo Gareca has not entirely ruled out the forward.
Gomez missed the last World Cup after an injury-plagued season but is hoping to convince Germany coach Joachim Loew to take him to Russia.
Cologne have 17 points, eight adrift of safety, while Stuttgart climbed to ninth with 33.
Bayern were gifted the lead at Freiburg by an Alexander Schwolow own goal in the 25th minute and Corentin Tolisso added another three minutes later.
Sandro Wagner and Thomas Mueller completed the rout in the second half.
(Writing by Brian Homewood in Bern, editing by Ed Osmond)
