A second-half strike from midfielder Barry Ferguson - his first goal for the club - was sufficient for City to clinch an away meeting with Everton in the next round.
But the home side were indebted to two fine saves from on-loan keeper Joe Hart for keeping them in the tie before Ferguson struck after 61 minutes.
McLeish said: "Is it a relief? I wouldn't say it was a relief but I'm happy to get the victory from a tough game when we weren't at our best.
"Once again it tells you a lot about the resilience of the players that we can come out on top when not firing on all cylinders - and it was a terrific goal.
"But I have to say that Forest played well and they can count themselves unlucky to be out of the Cup."
McLeish decided to field the same line-up which had held Manchester United to a 1-1 draw at St Andrew's at the weekend.
He said: "I wanted to keep the momentum going that we have at present so I kept the same team.
"The players have not been training recently because of the weather so it is better that they play rather than have two or three days off and lose their sharpness.
"It was important they played tonight with a view to Saturday's game against Portsmouth."