Everton captain Jagielka was unhappy with the response he got when he asked Atkinson why a critical City goal, after Raheem Sterling ran the ball out of play, was allowed to stand.
Atkinson and his officials missed the ball crossing the byline before Sterling pulled it back for Kevin De Bruyne to put City 2-1 ahead in the second leg at the Etihad Stadium.
That contentious 70th-minute goal levelled the tie on aggregate and City went on to progress 4-3 overall after Sergio Aguero compounded Everton's woes with a header to make it 3-1 six minutes later.
Asked if he spoke to the referee, Jagielka said: "I tried to
He told me that my defending was brilliant
So, it is difficult when things start going the wrong way and they become a little bit arrogant - but unfortunately that is the way it goes with some.
"If he made a decision wrong or he didn't see it, that is what happens, but sometimes when you go and speak to people and, in such a big game, get the answers you get it is that little bit more frustrating."
The incident was the latest in a series of controversies to have hampered Everton, who are stuck in mid-table in the Barclays Premier League, in recent weeks.
In the past month they have been on the wrong end of a disputed penalty against Stoke, an offside strike at Chelsea and a handball incident against Swansea, all of which contributed to defeats.
Jagielka said: "It was a great cutback and a great finish but the replays show it is not just an inch or so out
You expect the officials that are supposed to be of a high standard to spot that but this obviously wasn't one of those nights.
"It did happen quite quickly but the way things have gone in the last few games, people can't see offsides and people can't see the ball out of play
We are obviously feeling a bit annoyed at some of the decisions we have got recently."
Everton led 2-1 after the first leg and put themselves in sight of a final against rivals Liverpool next month as Ross Barkley extended their advantage with a superb strike.
But City quickly replied through Fernandinho and hit the woodwork twice through Aguero and David Silva before substitute De Bruyne struck.
De Bruyne ended the match on a stretcher with suspected medial ligament damage but was outstanding in his 23 minutes on the field, and it was his cross that set up Aguero's clincher.
Sterling, who will now face a Wembley reunion with the club he left in an acrimonious Â£49million deal last summer, had no qualms accepting victory, feeling he and City were due a little luck themselves.
Jesus Navas had a strong penalty claim turned down in the first leg at Goodison Park while Sterling himself was denied an injury-time spot-kick in the Premier League goalless draw against Everton a fortnight ago.
Sterling said: "We got the luck of the break, and we haven't had that in the last few games
I am just happy to get the good luck on this time, as the last few times we haven't, and it has taken us to Wembley, and that is the most important thing."
Sterling may get a rough ride from Liverpool fans but he is not concerned.
He said: "That is not a big thing on my mind
Obviously it is my ex team, but it is just another team on the way to get past.
"It (the final) is in my home city and it is something I have wanted to do since I was a kid
Nothing is going to stop me from doing it."
De Bruyne's knee was due to be assessed on Thursday but manager Manuel Pellegrini said he was "optimistic" the Belgian would not miss the rest of the season.
Source : PA