The reigning Premier League champions were already 2-0 ahead when Kompany was given his marching orders after Arsenal had played for much of the game with 10 men, with defender Laurent Koscielny dismissed for hauling down Dzeko inside the penalty area after only nine minutes.
Dzeko missed the resulting penalty but scored City's second following a terrific opening goal from James Milner.
City boss Roberto Mancini said immediately after the game that an appeal against the Kompany dismissal was likely as he did not feel Kompany had even fouled Wilshere.
"We will appeal, because it is impossible that we can lose one player for three games for nothing. It is not a red card, it is nothing," he said.
"He went into the tackle, anticipating the opponent and took the ball before.
"I don't know how it is possible to get a red card like this. It was no foul, no red card."
Kompany used Twitter to defend the challenge but stopped short of criticising Dean.
He wrote: "No grudges against the referee, I understand the difficulty of the job.
"About the tackle: If the ball is overrun by the opponent and a 50/50 challenge occurs, collision is inevitable.
"Ultimately I'm a defender: Appeal may work or not. I will never pull out of a challenge, as much as I will never intend to injury a player."
First-team coach David Platt believes the club have valid grounds to challenge referee Dean's original decision but also refused to lay any blame at the door of the referee.
"We feel as though there is a case," he said. "It is a difficult one for the referee because when you look at it live it certainly looks like what the referee would term as serious foul play," he told CityTV.
"I think when you slow the incident down you see that Vinny takes the ball and it is only one-footed. I think opinion will be divided.
"He certainly goes to ground but he actually takes the ball, I wouldn't blame the referee for his decision but there is doubt as to whether it is a red card.
"Of course it would be a loss (if the appeal failed), it is a loss when you lose any player and we have got Yaya Toure and Kolo away as well but we will just have to get on with it - that is what we have got the squad for."
If the appeal is unsuccessful and the three-match ban upheld City would be without their skipper for Premier League matches with Fulham and QPR and an FA Cup fourth-round tie against either Crystal Palace or Stoke.
Meanwhile Platt believes that beating Arsenal on Sunday afternoon was crucial if City were to avoid a 'significant gap' opening up between themselves and league-leaders Manchester United.
United's win against Liverpool earlier in the day had seen them move 10 points clear at the top of the table but Platt felt City's response was a good one.
"It was a terrific result," he said. "To come to the Emirates and get a victory and go away with three points I think is very, very pleasing.
"It is a difficult place to come to, particularly with the result that happened earlier in the afternoon as well, Manchester United would have been 10 points clear of us had we not taken the three points and that is a really significant gap."
One player who City could welcome back sooner rather than later is midfielder Jack Rodwell.
The 21-year-old England international moved to the Etihad from Everton over the summer but, after featuring regularly during the early stages of the season, he has not played since the 3-0 win over Sunderland on October 6 due to a number of hamstring problems.
But Rodwell is now around two weeks from a return and is looking forward to fully recovering from the lengthy injury.
"I feel really fit - I've been doing a lot of strengthening work because I've been out so long with my hamstring injuries so it's taken time," he said.
"My first injury was my left hamstring which kept me out for five weeks and then my right hamstring started to niggle which put me back to square one.
"It's easily the worst injury set-back of my career because I've been out for more than three months now."