Roberto Mancini will be pleased to have escaped with the points after another performance that will inevitably raise more questions over his side's lack of dynamism this season.
City can at least also reflect on a clean sheet - only the second of their season - after Joe Hart saved twice from Michu with the score at the Etihad Stadium still 0-0.
Tevez's strike briefly lifted the atmosphere but there was still plenty of anxiety around Eastlands during a game that ran to 12 minutes of injury time after lengthy stoppages for injuries to Michel Vorm and Micah Richards.
Michu did put the ball into the net with a fine finish but was well offside as he received the ball.
Swansea then pieced together a fine move which led to Jonathan de Guzman crossing for Michu, who powered a header straight at Hart. Richards sliced a clearance to give Ki Sung-yeung a follow-up chance but Kompany blocked.
City made the most of that reprieve and took the lead on the hour as Tevez took a pass from Gael Clichy in his stride and hit a swerving 25-yard effort past Vorm.
Swansea's woes were compounded by an injury to Vorm, who appeared to hurt himself stretching for the Tevez shot. He was unable to continue and left the field on a stretcher.
City suffered a blow when Richards, who missed the first two months of the season through injury, went down in agony holding his knee. He eventually had to leave the field on a stretcher wearing an oxygen mask.