The Cobblers created the better chances throughout and were good value for their victory but had to wait until midway through the second half to secure it.
Adebayo Akinfenwa, who headed against a post in the first half, nodded home Ben Tozer's cross to score his 15th goal of the season and propel Northampton into the play-off positions.
Bothroyd, who saw his team beaten 3-0 at Chesterfield last weekend, said: "We knew we had to improve after last week's display and we also knew that with hardly anyone else playing that we could give ourselves a nice psychological boost.
"We did that and we've still got a game in hand on some teams, so we're in a good position.
"Burton and ourselves are very evenly-matched. We have almost identical home records and almost identical away records and they were going to take some breaking down but I thought we got what we deserved and were worth our win.
"They had a game plan and are a good team. We knew we were going to have to keep some good players quiet and I thought we did that. We were mentally alert and we had to grind out the win, which we did."
Boothroyd also had words of praise for the club's supporters who helped to get the game on.
He added: "The fact that our supporters were here at 7.30am on the morning of the game put us under a little bit more pressure actually but they did their bit and we did our bit.
"We really appreciated their efforts and the players did too - I was asked by them to pass on their thanks. After they did that it was a case of us doing our job."