City's Robbie Fowler missed three great chances in 5 second half minutes, before Eidur Gudjohnsen finished off Wayne Bridge's through pass in the 82nd minute.
"It was a difficult match," said Ranieri. "City started well in the first and second half.
"We know City are good when they play well. They are a dangerous team, we knew that, so it was a good performance for us after the big battle in Stuttgart.
"We are a little tired but I am pleased with the performance.
"Luck at the end of the year evens itself out. If we were lucky today, it is because we had no luck in other games, but that is football."
Claudio claimed that at this stage of the season, results are far more important than the performance.
"It was difficult for us to maintain a clean sheet, but it was important to get the points today," the Tinkerman added.
"Now is not an important time to play well, being more important to get the three points."