The expected game of few chances ensued - Seasiders boss Paul Ince is still searching for his first goal after three games in charge - but City and Steven Davies had a goal ruled out for offside in the second half.
O'Driscoll had few complaints, though, with Robins keeper Tom Heaton reacting well to deny Kirk Broadfoot's header, after picking up his first point on the road since taking the Ashton Gate reins on January 14.
"I thought we were consistent from the first whistle to the last and I think in the Championship that's quite difficult," said O'Driscoll.
"It's another game. This is a treadmill in the Championship. You don't look too far ahead and you don't look too far behind you. It's onto the next game.
"The most important moments of the game were a corner and a throw. The rest of it you could probably forget.
"Heaton pulled off a great save against Barnsley last week and this is another one (from Broadfoot). He's managed to claw it out and that would have been the difference between the two teams.
"I couldn't see the Davies offside goal from where I was standing. The linesman had a pretty good afternoon, though, so I would imagine it came off somebody and if Davies was in an offside position then it's offside."