Stefano Pioli's Lazio powered past Cagliari to a 4-2 win on Monday, which took them up to third in the standings with 19 points from 10 matches, three behind local rivals Roma and six adrift of leaders Juventus.
And Klose, who struck twice in the victory against Cagliari - his second and third goals of the Serie A campaign, has warned the rest of Italy's top flight that Lazio will accept nothing less than European qualification this season.
"It's an important victory. We worked extremely hard in the first half and deserved [the win]," the German striker said.
"I've been at Lazio for three years and see the work we do on the pitch. We wanted third place and we have demonstrated this.
"We go game by game but we want to finish at least in the top five."
Klose's first goal came after a smart knock-down from Senad Lulic, while the veteran forward's second was a deflected strike from just inside the box to give Lazio a 3-0 lead at half-time.
While Cagliari hit back in the second half, reducing Lazio's lead to 3-2 by the 84th minute, Lazio secured their fifth win in six matches thanks to Ederson's stoppage-time goal.
For Klose, who has started three of Lazio's 10 games this season for eight total appearances, Monday's goals took him to three goals in as many matches but the 36-year-old refused to put too much pressure on Pioli to start him more often.
"Do I want to play more? Everybody wants to play but all of us can make our mark," he said.
"As always, the coach decides."
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