All three goals came in the second half against a Northampton side who were looking to knock out an League One team for the third time in this season's competition.
However, David Mooney struck twice to put Orient on their way and substitute Michael Symes scored from the penalty spot to seal their victory.
O's assistant boss Nugent said: "Since our victory over Shrewsbury everything seems to have clicked but we never really thought we were that far away before then.
"There were a few games earlier in the season when we weren't creating chances but once we started to create them - and take them - things have turned around.
"But Northampton were the better side in the first half and played with a certain style which was causing us problems. It took a great save from Jamie Jones to keep us in it in the first half when we were being penned in."
Nugent was also delighted to see Mooney among the goals in the absence of regular frontman Kevin Lisbie.
The O's head to Bury in the league on Saturday, and Nugent added: "David has hit form at the moment and is clinical in front of goal.
"But there are a lot of players hitting form at the same time and we'll have a few selection problems for Saturday."