Billy Sharp had earlier spurned a host of chances as O'Driscoll's side made light of a dreadful pitch which showed the scars of the recent bad weather that had seen the game twice called off.
Rovers managed to play their usual flowing game throughout but were left to celebrate a scruffy goal from a corner as they saw off a gutsy Brentford outfit.
O'Driscoll said: "We were terrific on Saturday against Watford and in a different way we were again tonight.
"I'd watched Brentford twice and they were impressive but our attitude was first-class on a difficult surface.
"We knew it would be a test for our players but we got on with it, worked extremely hard and we are delighted.
"I would have taken a draw before the game but it could have been done and dusted in the first five minutes.
"We should have been out of sight but I had a look at the players at half-time and their heads hadn't gone down and they ground it out in the end.
"The Cup is a good competition and every club wants to go as far as they can. Financially it always helps and we are no different.
"The goal was fortuitous but you need that every now and again.
"You lose a bit of the romance when you play a team who are in the same division as you but it's one of the best grounds in the country and we will be up for it."
Sharp should have had Doncaster ahead inside the first minute and could have had a hat-trick in the first half but Brentford boss Andy Scott felt his side deserved a second chance.
He said: "We were well worth a replay.
"We got beaten by a fortunate goal but I'm very proud of the way that we played.
"The lads were excellent and you wouldn't have known the gulf in league positions between the sides from that performance.
"We can take a lot of positives from the performance. They could have scored a few times in the first 10 minutes but after that we caused them problems.
"Defensively we looked solid and we coped with their movement which has won them a lot of plaudits.
"Now we are concentrating back on the league and if we play like that we will be okay."
Scott managed to extend the loan of Tottenham youngster John Bostock before kick-off and he revealed that he hopes to hold onto the 18-year-old for a further month.
He said: "We made it a week just to get him in the squad for this game and Saturday but we will look at that and hopefully get him for another month."