Klose pounced eight minutes from time to beat Samir Handanovic with a ruthless finish to give Vladimir Petkovic’s side all three points.
The Germany international's winner, his 10th goal of the season, lifts Lazio into fourth place on 33 points, level with Napoli, while second-placed Inter missed the chance to close the gap between them and Antonio Conte's champions Juventus to one point.
After a scrappy opening, the hosts crafted the first scoring opportunity of the evening in the third minute when Hernanes latched onto Senad Lulic and narrowly missed the top corner.
Lazio continued to see more of the ball in the first half, but neither side fashioned a clear-cut opportunity as the two defences dominated.
Fredy Guarin came agonisingly close to giving Inter the lead just after the hour mark as he struck the post with a rasping drive from the edge of the Lazio box with Federico Marchetti comprehensively beaten.
The Lazio goalkeeper made a stunning double save in the 69th minute as he tipped a fizzing shot from Antonio Cassano onto the post, before producing a brilliant reflex stop to deny Yuto Nagatomo from close range with the follow up.
Lazio ended the match strongly, and Klose was convinced that he had won a penalty for his side 11 minutes from time as he went down under a faint challenge from Walter Samuel, but the referee shook his head with disdain.
But three minutes later, the deadlock was finally broken: Klose raced through on goal after latching on to Stefano Mauri’s incisive defence-splitting pass and slid a clinical finish beyond Handanovic.
Inter threw everything at their hosts in the closing minutes, but could not muster a late equaliser as Lazio preserved their slender advantage and claimed all three points.
The result represents a real setback for Andrea Stramaccioni’s side, who have the chance to respond against Serie B side Verona in Coppa Italia action in midweek, but for Lazio it was a victory to enjoy.