With new owners and a new manager in O'Driscoll, Forest had to be patient in their attempts to break down a stubborn City side.
But Cox, signed in the last week from West Brom, was left on the bench following his appearance for the Republic of Ireland in Serbia in midweek but he made an instant impact on his arrival.
He latched onto an Andy Reid through ball and cut back from the by line for Adlene Guedioura to mark his first league game as a permanent Forest player by sidefooting home from eight yards.
"Simon only trained with us on Friday and he was very tired but I knew he had 20 minutes in him. Common sense dictated that he could make an impact at the end," O'Driscoll said.
"It was something of a calculated risk to send him on and while we thought a goal was going to come we felt we needed an extra push on what was a very hot day.
"Simon wants to play and he wants to score goals. It's a fact that we don't score enough goals and he's been brought in to try and change that statistic but he's made the winner today and that's great.
"Simon had a hand in winning the ball in the first place and the goal was good because it's not very often you are attacking the opposition goal having set up a situation where you have got five on three.
"It's a beginning. People say: 'This is a good start'. But a start is 10 games. You always want to get three points. It was a good beginning, it was always going to be tight.
"It was a typical opening day of the season - hot - I thought both sets of players dealt with the conditions."
City boss Derek McInnes was furious that some decisions went against his team, including a possible sending off of Reid and a penalty shout when Liam Fontaine went down under challenge from Danny Collins.
He said: "I thought Forest could have been down to 10 men when Reid reacted the way he did against Greg Cunningham, who was perfectly in his rights to play-on after the referee had insisted on a dropped ball.
"Reid ran and over and adopted an aggressive manner to Greg, pushed him in the throat and the chin and sent him to the ground. We have been told that kind of thing will be punished with a red card and if that wasn't an aggressive manner I don't know what is.
"I thought we could have had a penalty as well but decisions went against us today. We've got to hope they go our way in other tight games.
"What we have to do now is move on, take the positives from the game and try and get the three points against Crystal Palace on Tuesday night.
"As an away team we were not to open and for long spells we kept Forest at arm's length but credit to them for coming up with the goal. Having said that I thought we were worthy of a point."