Remarkably, the goals were Klose's first strikes of 2013 and his first hat-trick in a Lazio shirt, as the Eagles rediscovered their goalscoring touch against the hapless Rossoblu.
The Biancocelesti did the damage in 17 first half minutes, taking a 4-0 break into the interval before sealing the rout on the hour mark. They stay hot on the trail of Udinese and city rivals AS Roma for a place in next season's Europa League.
Coming into the game, Vladimir Petkovic's side had lost five of their last eight, so he altered his formation to a more adventurous 4-4-2. Stefano Pioli's Bologna had drawn their last five and have all but consolidated their Serie A status for another season.
Manolo Gabbiadini fired an early free-kick over for the visitors, but after that it was all one way traffic from the home side.
Klose scored his first goal since mid-December halfway through the first half. Antonio Candreva's shot was parried by Stojanovic straight into the German's path and he swept in the rebound from five yards.
Hernanes doubled the lead just past the half hour, curling a 30 yard shot in off the underside of the bar. Sergio Floccari tried to replicate the effort, but saw his shot ricochet off the woodwork.
Klose netted his second after tucking Senad Lulic's layoff under Stojanovic into the bottom corner. Three minutes later, the hat-trick was complete when Klose got on the end of Abdoulay Konko's ball across the box for a simple tap in at the far post.
Federico Marchetti saved on the line to stop Panagiotis Kone from stabbing in a consolation at the start of the second half, before Klose ended any hopes of a fightback. The veteran was left unmarked to head in a fourth from close range, following a Candreva cross from the byline.
Klose scored his fifth goal with his fifth shot on target on 61 minutes. Floccari and Lulic sprung a quick counter attack, with Floccari rolling across for the unmarked frontman to register his 15th strike of the season.
Lazio: Marchetti; Konko, Dias, Cana, Radu; Ledesma; Candreva, (Pereirinha 63) Floccari, (Gonzalez 75) Hernanes, Lulic; Klose (Saha 68)
Bologna: Stojanovic; Motta, Sorensen, Antonsson, Abero; Perez, (Krhin 53) Taider; (Guarente 46) Gabbiadini, Diamanti, Kone; Gilardino (Christodoulopoulos 53)
Goals: Klose 22, 36, 39, 50, 61 (L) Hernanes 32 (L)