Fly-half Ian Keatley kicked six penalties for the visitors and despite conceding their first try in 366 minutes of rugby to Alex Cuthbert's score, they held on to claim their fourth successive victory this season.
Cardiff had three tries ruled out but O'Driscoll put the result down to his side's effort and resilience.
He said: "I think each player made 15 or 20 tackles. That's phenomenal. It's hard work. But that's what we are, hard workers, and that's what got us the result tonight.
"It's about belief. It's about hard work. I don't mean to be cocky or arrogant. We're not a cocky side. We're not an arrogant side. We believe in ourselves and we believe in our coaches."
Next up is a trip to Murrayfield to face Edinburgh on Friday where they will be looking to extend their overall winning run, which stretches way back into last season, to 15 after coming through a tough encounter in Cardiff.
O'Driscoll added: "Remember we've being coming here and getting beaten. We won tonight, it was close, and we won last year. But before that I think it was 2005 the last time we won in Cardiff.
"We knew we wouldn't be coming here and scoring four tries. We knew we had to make the most of whatever possession they'd give us. They're a good side. They didn't give us a lot of ball.
"Of course having Keats there to kick the penalties was obviously massive and we probably copped a bit of luck at the end with that forward pass but maybe that was down to pressure.
"This is never an easy place to come to get a result. We set a target for ourselves and I have to be very happy with our efforts."
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