The Germany international's 82nd-minute winner, his 10th goal of the season, lifted Lazio into fourth place on 33 points, level with Napoli, while second-placed Inter missed the chance to close the gap on Antonio Conte's champions to one point.
"We put on a good performance and we deserved to have the lead [when Lazio scored], but that's football," Stramaccioni told reporters.
"We had a bit of bad luck over the course of the game. Klose is a champion and he scored a fantastic goal, but we definitely did not deserve lose... the way we played we deserved a result."
In Saturday's other game, Antonio Di Natale rescued a point for 10-man Udinese when he scored an 89th-minute equaliser against Palermo, leaving the Sicilians a point above the relegation zone.
Defenders dominated the first hour in Rome and the first clear scoring opportunity did not arrive until the 63rd minute, when Freddy Guarin, the best of Inter's disappointing attacking players, struck the inside of the post with a low long-range shot.
The match then burst into life with Antonio Cassano following up Guarin's effort by having his shot pushed on to the same post by Lazio keeper Federico Marchetti and Yuto Nagatomo failed to score from the rebound.
Inter had their tails up and Rodrigo Palacio curled another shot just wide.
With 11 minutes left Klose missed a great chance when he miscontrolled the ball on the penalty spot and let it bobble away to Samir Handanovic in the Inter goal.
He made up for that miss, though, with eight minutes remaining when he raced on to Stefano Mauri's intelligent through-ball and beat Handanovic with a powerful shot.
In Udine, Palermo led from the 33rd minute through Josep Ilicic, who finished off a swift break by cutting in from the right and hitting a shot that took a freak bounce over the already committed keeper Zeljko Brkic.
Despite Udinese playing a man down for half-an-hour following Thomas Heurtaux's red card for two bookable offences, Palermo could not make the game safe and missed several chances.
They paid for their profligacy a minute from time when Di Natale pounced after Palermo keeper Samir Ujkani spilled the ball right in front of the striker.
