Mancini won the Italian Cup six times as a player and four as a manager and, having overseen two league victories after replacing Mark Hughes at City, watched his new team advance comfortably with a late first-half goal by striker Benjani Mwaruwari against the toothless second division hosts.
There were no real upsets on Saturday, although Liverpool had to come from behind to draw 1-1 at Championship side Reading and Portsmouth, the Premier League's bottom team, were held 1-1 at home by Coventry City.
Last season's runners-up Everton beat Carlisle United 3-1 while Tottenham Hotspur eased past Peterborough 4-0.
Championship side Preston North End, who sacked manager Alan Irvine last week, thrashed Colchester United 7-0 with a hat-trick by Jon Parkin while Bolton Wanderers, also managerless after unloading Gary Megson, beat Lincoln City 4-0.
In two all-Premier League ties Aston Villa beat 10-man Blackburn Rovers 3-1 and Wigan Athletic defeated Hull City 4-1.
MINOR LEAGUE
There was a hat-trick of defeats for minor league sides as Barrow went down 3-0 at Sunderland, Luton Town lost 1-0 at Southampton and York City, having briefly led at Premier League Stoke City after arriving late due to traffic, lost 3-1.
Championship side Nottingham Forest, unbeaten in 16 games, unsurprisingly drew 0-0 with Birmingham City, undefeated in 11, with the home side's Rob Earnshaw missing a penalty.
If Mancini was hoping for a special atmosphere in his first taste of the FA Cup he would have been severely disappointed by a flat occasion that attracted only 12,474 to the Riverside.
Boro barely managed a shot on goal all afternoon while big-spending City were not much better.
The decisive goal was well-made though as Martin Petrov ran from his own half and timed his pass nicely for Benjani to mark his first start of the season with a confident finish.
Wembley might seem a long way away but Mancini can take heart from the fact that in the last five years the team who have knocked Middlesbrough out have reached the final -- as long as nobody tells him they have all finished as runners-up.
"I like cup football and I want to do well in the FA Cup as well as the Carling (League) Cup," Mancini told the club's website ahead of City's midweek semi-final first leg against Manchester United."
The Italian also praised youngsters Dedryck Boyata and Vladimir Weiss, who made their first starts.
"I have never been afraid of giving young players their chance," he said. "It's important to me, the owners and the fans that we continue to bring players through our academy."
Seven times winners Liverpool put a full-strength team out but it was Reading, fifth-from-bottom in the Championship with just one home league
