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 broke the deadlock after with a stunning strike from 25 yards after a dull first hour.
That 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.
Following criticism of his defensive formations after the midweek loss to Ajax, Mancini dropped Joleon Lescott and played Matija Nastasic in a flat back four.
The two teams effectively cancelled each other out in a first half that must rank as one of the least entertaining 45-minute spells seen at the ground in recent times.
There was little by way of goalmouth action and the hosts' uncertainty transmitted to their fans.
City's side featured two left-backs in Gael Clichy and Aleksandar Kolarov.
Kolarov played most of the first half on the left wing but did interchange with Clichy at times, suggesting a continued lack of ease with whatever system City were playing.
Kolarov made an early driving run but Samir Nasri was crowded out on the edge of the box.
Watch: Tevez strike gives Man City points... More Videos
One of Swansea's early attacks saw Angel Rangel go down under a challenge from Gareth Barry but referee Martin Atkinson booked him for diving.
As the game settled in to a slow tempo, Tevez tried to make something happen for City but his low cross was gathered by Vorm.
Swansea exploited some uncertainty between Vincent Kompany and Nastasic in the City defence as Rangel crossed from the byline but Richards came across to cut off Michu.
The Swans striker was injured in the challenge and Hart was very sporting to call for play to be halted while he received treatment.
City could have benefited from an error when Vorm gifted possession to Sergio Aguero but Tevez was unable to take up the attack.
Michu did put the ball into the net with a fine finish but was well offside as he received the ball.
Hart showed he was as alert as ever when Michu raced clear in the closing stages of the first half with a well-timed run, blocking bravely from the Spaniard.
The half-time whistle led to boos from some City fans after their side's uninspired display but they looked a different proposition immediately after the break.
Mancini sent on Mario Balotelli for the ineffective Kolarov and within two minutes the Italian had seen a shot blocked. Aguero also tried to find his range but fired wide.
Balotelli had a great opportunity as he latched onto a fine pass from Nasri and took the ball round Vorm, but the keeper did just enough to force him wide and the chance passed.
Balotelli was in the thick of the action again as Richards whipped in a cross and the forward went down, claiming he was pulled back by Ashley Williams. Atkinson was not interested and Williams gave Balotelli a mouthful.
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 Clichy in his stride and hit a swerving 25-yard effort past Vorm.
It was his first goal in eight appearances and lifted the atmosphere at Eastlands, with even a 'Poznan' breaking out.
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.
Despite taking the lead, City were still sluggish and De Guzman attempted to catch them out with a long-range effort but missed the target.
Yaya Toure tried to create something for the hosts after combining with Tevez but his cross avoided everyone.
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.
That led to a lengthy, nervy and very long period of injury time but City managed to hold out.