Podolski created Cologne's first goal for Milivoje Novakovic just before half time before scoring the second just after the interval.
Podolski's second and Cologne's third sealed the win in the 54th minute before Simon Rolfes grabbed a consolation for the home side.
Andre Schurrle was sent off late on before Mato Jajalo completed the scoring with Cologne's fourth deep in stoppage time.
Cologne coach Solbakken knew that a win in the derby would be the perfect way to get his club's fans on his side after a difficult start to the season, but for much of the first half, it looked like the pressure was only going to grow on the Norwegian.
Andre Schurrle tried all he could to inspire Leverkusen back to victory after their defeat at Chelsea in midweek.
But it was the visitors who drew first blood in a move which was started by Podolski inside his own half and ended with the Germany international squaring for Novakovic to score with a simple tap-in from six yards out.
The two Cologne forwards exchanged passes again in the 47th minute after Stefan Reinartz's error allowed Podolski to break free.
This time it was the German's turn to score into an open goal from six-yards out from Novakovic's cross.
Another mistake by Reinartz led to Cologne's third as Podolski raced past the Leverkusen defender and shot into the far corner of the goal.
Jajalo almost made it four in the 67th minute when he headed against the inside of the right-hand post at the joint with the crossbar, prompting more boos from a far from amused crowd.
They were given something to smile about three minutes later when Rolfes tapped in a consolation after Hanno Balitsch's header was only parried by Michael Rensing.
But it was Cologne who continued to pose the greater threat with Podolski going close to completing his hat-trick before Schurrle was shown a red card as frustration got the better of him.
Jajalo then completed the rout with both Podolski and Novakovic involved in the build-up to another simple tap-in before the referee brought an end to Leverkusen's suffering with the final whistle.