Edin Dzeko struck the winner in the 35th minute at the Etihad Stadium when he volleyed home from close range after Carlos Tevez’s effort rebounded off the crossbar.
Mancini, whose side take on Wigan Athletic in the FA Cup final on Saturday, made eight changes to his starting line-up.
And the Italian was thrilled with the victory, which guarantees City’s place in Europe’s elite club competition next season.
“It was a difficult game because we changed a lot of players but it was important for us to win the game,” Mancini said after watching his side keep their 17th clean sheet of the season.
“After 10 minutes we started to play better and we missed five incredible chances to score which made it difficult for us in the second half.”