Boro, whose backline had to cope with the absence of suspended centre-back David Wheater, got their only goal through a first-half Julio Arca penalty, converted after after Gabor Gyepes brought down Marvin Emnes in the box.
Mowbray said: "The players put in an amazing shift of 90 minutes and I am delighted.
"Cardiff are a quality side and with their extra man in midfield and their pace going forward we had to work very hard and we did. We never wavered, especially in the last 20 minutes when they piled on the pressure. Everyone did their jobs.
"It is nice to get the win but it is about fine margins. We have played well in recent games and not got the rewards. Since I came here apart from the 2-0 win at Scunthorpe there's not been a game with more than one goal in it. It's that type of league.
"It was close again today but we got our reward. Now we have to build on it."
Cardiff boss Dave Jones pointed his finger at his leaky defence after a disappointing defeat.
Jones said: "I don't know how often we are going to leave the door open at the back. A bit of fear has crept in there somehow. At the moment that unit isn't working.
"We are the second biggest scorers so there is nothing wrong going forward but if you don't score you have to keep a clean sheet and that isn't happening and too often we end up chasing the game.
"I wouldn't mind if we were beaten by good movement but it was a punt down the middle and a sloppily conceded penalty and that shouldn't happen.
"We have to make sure we don't get carried away and let a blame culture creep in but right now they need a bit of criticism. It is harsh but they have to accept it."