Boro went ahead early on through Lukas Jutkiewicz but Stanley clawed level before the break courtesy of Marcus Carver, and the League Two side then snatched it late on at the Riverside with a strike by Piero Mingoia.
"We have no excuses," said Boro boss Mowbary. "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 Mowbray insisted his side would bounce back quickly from the humbling defeat.
"They will feel the hurt of it like we all do but they are professional footballers and will quickly move on and focus on the next game," he added.
- Sports & Recreation
- Lukas Jutkiewicz
- Marcus Carver