Premier League City were rarely troubled by Gordon Strachan's team and they booked their place in the fourth round with ease ahead of Wednesday's Carling Cup semi-final first-leg at home to Manchester United.
With the unexpectedly severe weather conditions in the north of England leaving a heavy covering of snow on Teesside, this tie was in doubt until referee Stuart Attwell sanctioned the fixture to go ahead as planned following an inspection of both the playing surface, and the roads surrounding the stadium, ninety minutes before kick-off.
A number of coaches carrying City supporters to the north-east were forced to turn back to Manchester, however, due to treacherous road conditions across the Pennines.
Throw in the elements and the unfamiliar look of Mancini's starting eleven and Boro, relegated from the Premier League last season, would have known that they had a potential upset within their grasp.
A lengthening injury list, plus the African Cup of Nations commitments of captain Kolo Toure and Emmanuel Adebayor, had left Mancini with few options, but with the Carling Cup clash again United looming on Wednesday, the Italian chose to err on the side of caution by starting with the likes of Carlos Tevez, Gareth Barry and Craig Bellamy on the bench.
Belgian youngster Dedryck Boyata was handed his debut at centre-half and Zimbabwean forward Benjani made his first start of the season up-front in an unfamiliar City XI.
Middlesbrough, who went into this game having won just twice in ten games since Strachan's arrival in October, saw their hopes of a shock recede dramatically when winger Adam Johnson limped out with a recurrence of a hamstring injury before the interval. Without Johnson, Boro lacked the one player capable of worrying City.
Mancini's team were rarely troubled by the home side in the first-half and City should have gone in with a greater half-time lead than the one goal provided by Benjani in the 45th minute.
Argentine full-back Pablo Zabaleta was unlucky to see a goalbound shot deflected behind by Julio Arca on six minutes, while Martin Petrov wastefully shot wide from 12 yards after being released by Benjani on 39 minutes.
Benjani, whose City career has been disrupted by a series of injuries since his arrival from Portsmouth in Jan 2008, rewarded Mancini's team for the dominance, however, by scoring the opening seconds before half-time.
An astute pass by Petrov found Benjani on the edge of the Boro penalty area and the forward beat goalkeeper Danny Coyne with a crisp right-foot strike. City were more comfortable than the 1-0 scoreline suggested and Middlesbrough's desperation was borne out on 47 minutes when captain Gary O'Neil was booked for diving by Attwell when he attempted to win a penalty following a challenge by Vladimir Weiss.
If any team was going to add to the scoring, City appeared most likely and substitute Tevez twice went close to doubling the visitors' lead.
Middlesbrough were restricted to set-pieces as they attempted to equalise, but both Jonathan Franks and David Wheater failed to take advantage of good headed chances as the game drifted towards the closing stages.
Tevez then sent a free-kick narrowly wide from 30 yards as City saw out with the final ten minutes without concern.