Roberto Mancini will have been 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, although they at least kept only their second clean sheet of the season after Joe Hart saved twice from Michu with the score still 0-0.
Tevez broke the deadlock on the hour with a stunning strike from 25 yards. That briefly lifted the atmosphere but there was still plenty of anxiety around Eastlands during a game that ran to 12 minutes of added time after lengthy stoppages for injuries to Michel Vorm and Micah Richards.
Michu had the ball in the net late in the first half but the flag was up for offside. The Spaniard then rushed clear with a well-timed run, but Hart blocked bravely.
The half-time whistle was greeted with boos from some City fans after their side's uninspired display but they looked a different proposition immediately after the break.
Roberto Mancini sent on Mario Balotelli for the ineffective Aleksandar Kolarov and within two minutes the Italian had a shot blocked. Sergio Aguero also tried to find his range but fired wide. Balotelli had a couple of further opportunities before, at the other end, Michu powered a header straight at Hart and Vincent Kompany blocked Ki Sung-Yueng's follow-up.
City made the most of the reprieve when Tevez took a Gael Clichy pass in his stride and hit a swerving effort past Vorm.
The goalkeeper appeared to hurt himself stretching for the shot and was unable to continue, leaving the field on a stretcher.
City suffered a blow of their own when Richards, who missed the first two months of the season, went down in agony holding his knee. He eventually had to be stretchered off wearing an oxygen mask. That led to a lengthy and nervy and period of injury time but City held out.