The 25-year-old Bosnian's sublime 22nd minute header proved enough to sink the bottom club after Wigan goalkeeper Ali Al Habsi had threatened to frustrate the leaders.
Al Habsi produced a string of excellent saves both before and after Dzeko had netted his 14th goal for City this season.
David Silva, Sergio Aguero and Dzeko were all denied by the Oman international but City returned to winning ways to pull clear of nearest rivals Manchester United ahead of Sunday's big home game with third-placed Tottenham.
"I could have scored one or two more goals in the end, but I'm happy with the three points because it was a tough game and Wigan played very well," Dzeko told Sky Sports.
"I'm happy to score and help my team win. In football sometimes you score and sometimes not, but I'm happy today."
Having suffered back-to-back defeats in cup competition, an away game against basement team Wigan provided City with a perfect opportunity to snap out of their winter blues.
The FA Cup elimination at the hands of Manchester United as well as the League Cup first leg home defeat to Liverpool, together with captain Vincent Kompany's four-match suspension, had resulted in a poor start to 2012 for Mancini's expensively-assembled side.
But they found relief at Wigan, whose record against the so-called big four is nothing short of diabolical.
Since winning promotion to the top-flight in 2005, Wigan have lost all 14 league and cup games against Manchester United while they have now gone eight hours 15 minutes without scoring against City.
Mancini was relieved to have stopped the rot.
"It was about the victory not the performance tonight," he said.
"It was really important to win tonight. We knew that was difficult because Wigan this month have played very well and Roberto Martinez is a fantastic manager."
Martinez for his part was annoyed by their first-half performance but buoyed by the spirit shown in the second period.
"The manner that we finished the game is probably what pleases me most," said Martinez, whose side only escaped relegation on the last day of last season.
"We probably had our worst first half this season, we were cagey, but in the second half we took control a bit more and I had the feeling in the last 20 minutes that there could be a chance coming our way."
Wigan started brightly with Scottish midfielder James McArthur seeing a third minute shot deflected narrowly wide by James Milner while the hosts were also denied a sixth minute penalty.
Martin Atkinson has sent off six players this season but the referee did not want to know when City's England midfielder Gareth Barry appeared to lean into Jordi Gomez deep inside the visitors penalty area.
But Wigan, like their season, quickly fizzled out and City, captained by Pablo Zabaleta on the Argentinian's 27th birthday, gradually began to take command.
The home side, who had conceded 11 times in their previous three league games, were undone by a simple set-piece midway through the first half as Dzeko found the net with a superb header after meeting Silva's inch-perfect free-kick.
It was the former Wolfsburg player's first goal in 14 games for club and country yet despite the quality of the finish, it was far too easy for City.
Wigan are now without a win in their last nine home attempts in the league yet they can count themselves fortunate to have escaped a hiding.
Al Habsi produced a flying world-class save to deny Dzeko a second from 12-yards after 55 minutes before the Oman international frustrated Silva moments later.
Then Aguero went on a sensational mazy run which had Wigan's defence chasing shadows.
But just when the former Atletico Madrid player looked set to walk the ball into the net, Dzeko spoiled it all by taking the ball off his toe and allowed Al Habsi to make an easy save.
"I was waiting for him to shoot," explained Dzeko.
"In the end the ball came to me and he wanted to shoot, and I wanted to shoot, but it doesn't matter. We're happy now."