Mancini said the champions would appeal referee Mike Dean's 75th-minute decision in Sunday's 2-0 victory, which followed the Belgian centre-back's tackle on Jack Wilshere where he won the ball but with two feet just off the ground.
"We do appeal for Vincent. We have problems because we are missing lots of players," Mancini told reporters, saying it was a one-footed tackle where he anticipated the player and took the ball.
"It is impossible that a player misses three games for nothing.
"It was not a foul, no red card."
With Sergio Aguero, Yaya Toure and Samir Nasri all missing, City's success was all the more satisfying for Mancini even if they were given a large dollop of luck when Arsenal's French centre-back Laurent Koscielny was sent off after 10 minutes.
"The rules are last man," he said of Koscielny grabbing Edin Dzeko around the waist in the box and dragging him to the ground.
Bosnia striker Dzeko missed the resulting penalty, managing to hit the goalkeeper, post and goalkeeper again from one shot but City were soon ahead through James Milner's excellent 21st minute strike.
Reports before kick-off said City had returned 900 of their 3,000 tickets after fans baulked at Arsenal's 62 pound price. Tickets for future high profile games at City's Etihad Stadium were available on Sunday from around 46 pounds.
"It was an important victory. We knew it was difficult to play here," Mancini added.
"United is a strong team and in this moment they are playing very well but the season is long.
"I don't think we deserve to be seven points behind."
