The Germany international netted his hat-trick before half-time then came out firing again after the break to put Bologna to the sword in a dominant performance.
Hernanes also got onto the scoresheet for the home side with their second goal of the day, before Klose stole all of the headlines with a superb display.
Lazio, who are now three points adrift of fifth-placed Roma in the final European spot, started the game on the front foot and missed several good chances before Klose's opening goal in the 22nd minute.
Sergio Floccari flashed the ball wide from 30 yards early on before Christian Ledesma and Klose both put the Bologna defence under pressure.
Antonio Candreva was unlucky not to break the deadlock just after the quarter-hour mark but he quickly turned provider, as his shot was saved by goalkeeper Dejan Stojanovic minutes later, with Klose on hand to slot in the rebound.
Lazio doubled their lead 10 minutes later through Hernanes' curling shot from the edge of the box before Floccari almost made it three when his powerful strike hit the crossbar.
Klose then scored twice in three minutes to secure his hat-trick before the half-time whistle, first capitalising on Senad Lulic's counter-attack before beating Stojanovic one-on-one as he got on the end of Abdoulay Konko's long cross.
Bologna squandered two good chances to gain a foothold in the game before the break when Saphir Taider missed twice, and they were soon back on the defensive when Klose fired his fourth goal five minutes into the second half.
The German striker was unmarked in the box and headed Candreva's cross into the net as Lazio underlined their grip on the game.
The sixth goal, and Klose's fifth, came in the 61st minute when he linked up with Floccari, who provided a perfect ball in, to slot the ball home.
Bologna rallied as they bid to regain some pride, but they were unable to beat the defences of Lazio keeper Federico Marchetti.