Dzeko, a 27 million pounds buy from VfL Wolfsburg, made a good first impression as City led 4-1 but they were hanging on at the end against the doughty visitors.
Third-placed Arsenal kept up the pressure with a 3-0 victory at West Ham United in a what, according to local media reports, could be Avram Grant's last game in charge of the bottom club.
Chelsea were untroubled in beating Blackburn Rovers 2-0 with second-half goals by Branislav Ivanovic and Nicolas Anelka.
West Bromwich Albion beat Blackpool 3-2, Stoke City beat Bolton Wanderers 2-0 and Wigan Athletic drew 1-1 with Fulham.
Wolves, who beat City at Molineux in October and have taken the scalps of Liverpool and Chelsea in recent weeks, took a deserved lead after 12 minutes through Nenad Milijas.
City rode their luck and levelled five minutes before halftime through Kolo Toure and went ahead four minutes after the restart when Tevez skipped past three challenges and finished superbly.
Dzeko set up Yaya Toure for a simple finish after 54 minutes and Tevez headed the fourth after 66, his 14th league goal of the season.
Wolves though hit back with a Kevin Doyle penalty and after Ronald Zubar headed their third four minutes from time City were hanging on desperately.
"I was worried in the last eight minutes as we conceded a lot of chances and we must improve in this situation," City boss Roberto Mancini told BBC Radio 5-Live.
STAY ON TOP
"We played good football in the second half but we had some problems at the back and shouldn't concede three goals at home," Mancini told the club's website.
"But we are happy we can stay top for one day. Last time it was a few hours and maybe in May it will be for a long time," added the Italian, who praised Tevez for his "fantastic goals."
Chelsea have also looked a long way from title material in recent weeks with a return of 10 points from 11 games and there was relief all round Stamford Bridge when Ivanovic forced in a low shot after 57 minutes.
Anelka then diverted a goalbound Ivanovic header for the clincher after 76, his first league goal since October 30.
"We deserved it, we maintained good control for all the game but it was not easy because we took a long time to score the first goal," Chelsea manager Carlo Ancelotti told reporters.
"I hope the bad moment is over now. We are not at the same level but we will come back to that level."
In a muted atmosphere at West Ham, Arsenal took control from the start and went ahead after 13 minutes when Robin van Persie lashed in a shot from the edge of the box.
Four minutes before half-time the Dutchman crossed for Theo Walcott to bang in the second. After Wayne Bridge completed a hapless debut by fouling Walcott, Van Persie completed the cr
