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 but we got our reward. Now we have to build on it."
Source: PA
Source: PA