All three goals came in the second half of the Southern Area tie against a Cobblers side who were looking to knock out a League One outfit for the third time in this season's competition, having already seen off MK Dons and Colchester.
However, David Mooney struck twice to put Orient on their way and substitute Michael Symes scored from the penalty spot to confirm their place in the last four.
Boothroyd said: "It was one of those games when you look at a 3-0 scoreline and wonder how that happened.
"We were by far the better side in the first half and created a couple of really good chances. Their goalkeeper has kept them in the game and if anyone was going to score it was going to be Louis Moult, he's right up there at the moment for goals, but unfortunately he didn't have his shooting boots on.
"I was very pleased with our first-half performance but in the second half we had a couple of decisions go against us. As a manager you prepare well, the players work hard all week and I put out a strong side so it was disappointing to get decisions like that.
"I couldn't see where the free-kick came from for their first goal, although we should have defended it better, and for the handball for the penalty there was nothing Joe Widdowson could do to get out of the way of it. But you just have to deal with it.
"That's the way it goes, the Johnstone's Paint Trophy will have to wait for another year, and we'll just have to pick ourselves up and get back to the league."