Lukas Jutkiewicz fired the Championship side ahead inside the opening 10 minutes before Marcus Carver levelled matters on 40 with Piero Mingoia snatching Accrington's winner eight minutes from time.
A disappointed Mowbray said: "We are out because we didn't take our opportunities and they took theirs.
"And there were plenty of opportunities. The bottom line is you have to stick your chances away and we didn't do that. You are only as good as your strikers.
"You can't have that much domination in a match and not score but we will quickly move on and focus on the next game."
His opposite number at Accrington, James Beattie, added: "They threw everything at us and we put in a magnificent performance. I'm proud of them.
"We rode our luck at times, there's no denying that. They had a lot of chances and we did well. We put our bodies on the line. We showed commitment.
"There were some real heroic performances out there. I'm not going to single individuals out but they know who they are.
"It was a massive win for the club, financially it could be a great boost for us, especially if we get a lucrative draw in the next round."