Germany striker Klose struck twice in the space of seven first half minutes, relieving the pressure on coach Vladimir Petkovic who had been facing the sack after six league games without a win.
Fiorentina took fourth spot with a comfortable 3-0 defeat of Bologna in which Giuseppe Rossi scored his 13th goal of the season.
Carlos Tevez hat-trick had put leaders Juventus six points clear with a 4-0 rout of Sassuolo on Sunday
Tevez struck for Juve with simple finishes in the 15th, 45th and 68th minutes to move Antonio Conte's side up to 43 points, six ahead of AS Roma.
The 29-year-old also set up Federico Peluso's first Juventus goal in the 28th minute, sending in a beautiful free kick that the midfielder headed home with the help of Sassuolo defender Luca Antei and was given a standard ovation when substituted shortly after netting his third.
"Clearly there is disappointment at going out of the Champions League, but this has to give us more determination and motivation, because we were eliminated and did not deserve it," Conte said.
"I was at Formello (the team training ground) for three days and I realised that the team needed a win to get going again," he said. "I don't think that we are now in the position where we should fire the coach."
Genoa failed to build on their recent good run under Gian Piero Gasperini, dropping points at home when an injury-time header from Giuseppe De Luca earned Atalanta a 1-1 draw after Andrea Bertolacci had put the hosts in front.
A goal from Eder early in the first half moved Sampdoria four points from the drop zone with a 1-0 win at relegation rivals Chievo while Parma drew 0-0 with Cagliari.