It looked like the Teessiders had done enough when Gary O'Neil put them ahead after 58 minutes, but Tony Mowbray's side were stunned by two late goals from Shaun Harrad which saw them eliminated from the competition.
Peschisolido said: "I feel fantastic, but if you'd have asked going into the last 10 minutes of the game I wouldn't have said so. Obviously, I'm absolutely overjoyed that we've got through. It's a fantastic result for the club and the fans."
He added: "I actually thought we were terrible. We just didn't perform.
"Defensively we were solid, but on the ball it was a shambles at times. I'm amazed we've won the game, and maybe this will be a turning point for our season.
"Perhaps we didn't deserve it, but I'm not complaining because we're in the hat for the next round - and that's unbelievable.
"I'm going to enjoy my afternoon on Sunday looking at the draw. Let's hope we get a big one. It's massive for us to get through to the fourth round for the first time in the club's history. I'm pleased for the lads because they're an honest bunch.
"Middlesbrough had the lion's share of the possession and we probably shouldn't have won the game. But that's the FA Cup for you. It's a leveller and strange things happen and fortunately for us we came out on the right side."
Middlesbrough manager Tony Mowbray rued his side's inability to cope with pressure for a full 90 minutes. He said: "Last-minute goals are killing us. It happened at Elland Road last weekend and it's happened again.
"If we could have taken a better share of our chances it wouldn't really have mattered."