David Mooney, Elliott Omozusi, Gary Sawyer and Lloyd James netted their spot-kicks for the home side and although Leon Clarke and Jordan Clarke were on target in reply for the Sky Blues, Franck Moussa had his effort saved by Jamie Jones while John Fleck missed the target.
It ensured the O's triumphed 4-2 after their second-round tie at Brisbane Road finished goalless after 90 minutes.
"Make no mistake, Coventry are a good side," Nugent said.
"They moved the ball around well and I thought our discipline in contending with that was fantastic. We had to keep a really good shape and to be honest, we created the best chances in the game.
"Coventry made it difficult for us because they had a lot of possession.
"We made five changes from Saturday's game and it was good to give other members of the squad a game. We've got a few players who haven't had a lot of competitive games, only a few friendlies, but they showed a good fitness and kept well disciplined throughout the game.
"Those players who came in showed we have depth in the squad and that there is real competition for places. It's been difficult for the players who have been in the side because they've had to be patient as we've been doing so well."
Coventry goalkeeper Joe Murphy made second-half saves from Mooney and Moses Odubajo but, overall, it was of little surprise that a dour game needed a penalty shootout to decide the winner.
Nugent added: "We practice penalties quite often, particularly if we have a game in training that ends in a draw.
"All our players took good penalties so that was satisfying."