Boro were the better side for long spells at the Liberty Stadium but were unable to make it count on the scoreboard.
And that inability to convert chances into scores cost them dear when Seb Hines headed into his own net with nine minutes remaining.
"Cup competitions are about getting through to the next round, and we didn't manage that," said Mowbray.
"It was a tough game, fine margins. I could easily be sitting here having won 1-0, or having lost 1-0.
"Credit to Swansea, they are in the semi-finals of the cup and good luck to them.
"We gave a decent account of ourselves. It was a game we could have won. I don't think there was any time when they cut us open.
"I think the team can take some heart that if we can get out of this division we are in, we can come into the Premier League and be positive.
"There is no respite for us, really. We will try to do a session in the morning before travelling back, then we have one day's preparation and go again against Wolves, who were obviously a Premier League side last year.
"It's a big ask, but we keep going."