Kiessling opened the scoring in the sixth minute and then made it 2-0 after 32 minutes.
Robert Mak pulled a goal back for after 58 minutes, but Andre Schurrle restored the visitors' two-goal lead in the 77th minute and Kiessling completed his treble a minute from time.
Leverkusen finished the season in fifth place with 54 points, good enough for European football, while Nuremberg ended in 10th.
The hosts went close in the first minute as Daniel Didavi crossed from the right and Javier Pinola shot over the bar.
But Leverkusen took the lead after six minutes with Stefan Reinartz feeding a through-ball to Kiessling, who beat goalkeeper Raphael Schafer from 12 yards.
The away side nearly made it 2-0 as Pinola cleared a pass from the left side just wide of his own goal.
Leverkusen doubled their advantage in the 32nd minute with Michael Ballack feeding Schurrle, who passed the ball past Schafer to Kiessling, who slotted into the empty goal.
Nuremberg halved the deficit as Albert Bunjaku fed Mak to level.
But any hopes of a comeback were dashed in the 77th minute as Schurrle got Leverkusen's third.
Reinartz got down the left and crossed for Schurrle to convert.
And Kiessling completed his hat-trick to round off the scoring in the 89th minute, getting on the end of a long ball from Tranquillo Barnetta and guiding the ball between the legs of Schafer.