The veteran German striker tapped home a rebound after Julian Speroni had saved an Antonio Candreva strike in the 12th minute to ensure the Eagles will go into the new season on the back of a defeat.
Stephen Dobbie was among the bright sparks for the hosts in what was a quiet start to the match, with the close-season signing forcing two good chances for his side early on.
However, the crucial breakthrough came shortly after when 35-year-old Germany international Klose showcased his lethal touch in the box to put the Coppa Italia winners ahead.
Senad Lulic almost made it two after Candreva again forced a good saved from Speroni, but the Bosnian was unable to force the ball home.
Palace were denied a penalty before half-time when Dobbie went down in the box, while Jerome Thomas was forced off with an ankle injury.
The home side came close to an equaliser after 66 minutes when close-season addition Dwight Gayle played in fellow new recruit Kevin Phillips.
However, last year's play-off final hero shot straight at Federico Marchetti, leaving Palace boss Ian Holloway with questions to answer ahead of their clash with Tottenham on August 18.