The victory at Hinkle Fieldhouse was the ninth in a row for the Bulldogs.
Dunham provided a spark for a Butler squad that started slowly against the deliberate offense and pressure defense of the Privateers.
"We weren't playing with pace," Butler coach Brad Stevens said in a release. "We weren't playing in a stance; we weren't playing with connected screening."
He hit both of Butler's first half three-point field goals, while helping the Bulldogs to a 24-18 halftime lead, and then added nine more points in the second half, when Butler stretched the margin to double-figures.
Marshall gave the No. 17-ranked Bulldogs a solid inside scoring presence, while leading both teams in rebounds. He hit five of eight shots in the game, his third straight game hitting better than 60 percent from the field.
The Bulldogs (12-2) also picked up 11 points and six rebounds from sophomore Roosevelt Jones, and nine points and five rebounds from senior Andrew Smith.
The win came despite the fact that Butler's leading scorer, Rotnei Clarke, finished with a season-low four points.
"We just weren't very good," Stevens said. "You're going to have games like this throughout the course of the season, and I hope this is one of the very, very few. If you are going to have a game like this, it's good that you win it.".
Butler, ranked fourth in the Atlantic 10 Conference in scoring offense (74.0), managed just 10 points in the first 14 minutes of the game. Dunham's second three-pointer gave the Bulldogs a 13-11 lead with 6:03 left in the opening period and sparked a 14-7 Butler run to close the half.
The Bulldogs then opened the second half with a three-point field goal by sophomore Alex Barlow and a dunk by Marshall to stretch the margin to double-figures, 29-18.
New Orleans (3-10) never got closer than eight points the rest of the game.
Butler led by double-figures for the final 10 minutes of the game, with the biggest lead coming at 57-41, on a lay-up by Jones in the final minute.
The Bulldogs finished with a season-low 17 field goals, but managed 18 of 23 from the free throw line.
Butler finished with a 33-22 rebounding advantage and out-scored the Privateers, 19-2, on second chance points.
New Orleans, playing its fifth straight road game, used a very deliberate pace in the game, going deep into the shot clock on each possession before looking for a shot.
"It was one of those games where you were going to have to maximize your opportunities, because there weren't going to be very many of them," Stevens said.
New Orleans was led by Lovell Cook with 11 points, while Traddarius McPhearson added 10.
Butler will play its first Atlantic 10 game in program history as it travels to Saint Joseph's on Wednesday at 7 p.m.