Jordan Rhodes got both goals, scoring either side of the break, meaning the Robins have now lost their last four away games in all competitions.
Rovers have also beaten City three times this season, and completed their first ever league double over the Robins.
O'Driscoll said: "In this division set-plays play a massive part to concede from a second phase from a corner is disappointing but apart from that at half-time, we felt okay.
"I thought we could score a goal and were terrific at the start of the second half, but Danny Murphy hauls a ball back and we're two down."
O'Driscoll added: "It's the first time (since I've been here) we've had this sort of problem to solve but felt we went about it the wrong way.
"Lots of endeavour but we lost our shape and structure and you can't do that in this division, but for an hour I thought we were terrific."