After a 2-2 stalemate, Brighton won the penalty shootout when Michel Kuipers saved from Michael Ball allowing Matt Richards to seal the win.
Although City had rested their big-name buys Robinho and Shaun Wright-Phillips, and had their second choice keeper Kasper Schmeichel in goal, they had still named a strong side.
Following their takeover by Abu Dhabi Media Group, a consortium led by Sheikh Mansour bin Zayed Al Nayhan, a member of the Abu Dhabi royal family, City now have the deepest pockets in world football.
And whether they like it or not, exiting a competition to a team from two divisions below them will earn them negative headlines around the world.
"My lads were well up for it and we needed to do something big after losing to Walsall's nine men on Saturday," said Adams.
"But I'm not going to crow about this because as a player and a manager I've been on the wrong end of these type of results and I know how it hurts.
"But all I can say is that my boys were spot-on for their endeavour and their pride and deserve great credit. There was no pressure on us but we certainly didn't want to embarrass ourselves again.
"I'm sure we surprised City. They had some terrific players out there but I think our shape and determination surprised them.
"I had a feeling before the start we would do something but what a response after the reception we got for our performance on Saturday when we tried all we could to score against nine men but couldn't.
"The players were hurting from that still and determined to give the fans something back."