Mancini rested some of his big-name stars at the Riverside Stadium with a view to the Carling Cup semi-final against Manchester United next Wednesday.
Carlos Tevez, Craig Bellamy and Gareth Barry were all on the substitutes' bench.
Middlesbrough manager Gordon Strachan did not have the luxury of leaving out star players because of injuries, suspension and the departure of loan players.
Marcus Bent in fact was given the role of lone striker.
There was a sparse crowd on a bitterly cold day with reports of many City fans having been turned back by the police because of atrocious road conditions.
Snow, a swirling wind and freezing temperatures affected the standard of play.
Middlesbrough put up a spirited first-half performance without causing any problems for the visitors.
Adam Johnson showed flashes of skill so it was a tremendous blow for Boro when the England Under-21 international suffered a recurrence of a hamstring injury and was forced to leave the field.
It was remarkable in a low-key opening 45 minutes that neither side had a serious shot at goal. It was in first-half stoppage time that City went ahead with a Benjani shot from the edge of the penalty area. Martin Petrov was the architect with a tremendous run and a brilliant pass.
City brought on two of their big guns, Tevez and Barry, for the second half. Mancini was not wanting the game to slip from their grasp.
Two minutes after the break Boro skipper Gary O'Neil was shown a yellow card for diving. Shortly afterwards Strachan was forced to make a change with Marvin Emnes, a first-half substitute, being forced to leave the field through injury with Mark Yeates coming on to replace him.
The visitors were showing much more confidence and composure, and Boro were struggling to make any impact in the final third of the pitch.
With 20 minutes to go there was an historic FA Cup moment for Middlesbrough when 16-year-old substitute Luke Williams came on to the pitch.
In the previous match the local teenager had become the youngest player for 105 years to make his League debut for the club.
Bellamy came on for the final stages of the game as City booked a fourth round ticket.