Lukas Jutkiewicz gave Boro a first-half lead but was one of the main culprits as Mowbray's men passed up numerous chances to inflict further damage on their hosts.
At the other end, a clinical double from Luciano Becchio proved to be the difference.
"We had a host of chances and missed them," said Mowbray.
"We were through one-on-one two or three times and threatened the goal more than enough to have got something out of the game, but give Leeds credit.
"They worked hard and defended well without putting us under much pressure.
"They had a tough game against Chelsea on Wednesday and sent just about the same players out against us. They got a result so good luck to them.
"Becchio's first goal was a great one. To turn and hit the ball as well as that showed great technique, but our own finishing needed to be more clinical."