Klose's extraordinary performance began when the 34-year-old Germany international opened the scoring in the 22nd minute although there were suspicions of offside.
Hernanes scored a brilliant second with a dipping 30-metre shot, then Klose grabbed two in three minutes to complete a first-half hat-trick, finishing off a counter-attack and then tapping in Abdoullay Konko's pass at the far post.
Klose was allowed a free header from Antonio Candreva's cross as he added a fourth goal early in the second half and he was again unmarked as he turned in a Sergio Floccari pass just after the hour.
Third-placed AC Milan edged closer to the Champions League play-off spot when a late Mario Balotelli goal, his ninth since he joined from Manchester City at the end of January, gave them a 1-0 win over Torino.
Bottom club Pescara were relegated after one season in the top flight when they lost 4-1 at fellow strugglers Genoa with Marco Boriello scoring twice.
Argentine forward Gonzalo Bergessio got a hat-trick to give Catania a 3-0 win over Siena, pushing their opponents closer to Serie B.
They moved four points clear of Fiorentina who lost 1-0 at home to AS Roma on Saturday.
At the bottom, Palermo dropped back into the relegation zone with 32 points, swapping places with Genoa who have 35 while Torino are not out of the woods on 36.
Siena, 19th in the 20-team table, have 30 and Pescara 22.