"I'm absolutely overjoyed. It's a fantastic result for the club and the fans," said Pilgrims boss Peschisolido.
"I feel fantastic, but if you'd have asked going into the last 10 minutes of the game I wouldn't have said so. We were terrible and didn't perform."
Boro dominated the game and Gary O'Neil's superb 25-yard free-kick just before the hour mark looked set to put the Championship side into round four.
But Harad turned in a Russell Penn cross from 12 yards in the 82nd minute before smashing in the winner deep into four minutes of injury time.
"I'm amazed we've won the game," Peschisolido continued.
Well done guys. Skin of the teeth perhaps, but the record books will show for all time that on this day in history, Burton Albion finally reached the fourth round of the FA Cup
steve1210
"Defensively we were solid, but on the ball it was a shambles at times.
"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.
"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."
Boro boss Tony Mowbray was left to rue a host of missed chances and his side's inability to close out the game.
"Last minute goals are killing us," he said.
"It happened at Elland Road [against Leeds] last weekend and it's happened again.
"If we could have taken a better share of our chances it wouldn't really have mattered.
"They wouldn't have gambled like they did because they would have been two or three goals down and out of it.
"But when a team is only one goal down, they'll always put you under pressure in the last few minutes.
"It's another example of one goal never being enough.
"I don't even think it was a case of them putting us under any sort of pressure. It was just two isolated incidents. Even for the seven or eight minutes between their two goals there was only one team trying to score a winner - and that was us.
"Then they get a free kick in the 93rd minute and they scored from it. It was a case of two shots and two goals for Burton."
Source: BBC Sport
Source: BBC Sport