The home side extended their lead with a fine free-kick from Garreth O'Connor and secured all three points thanks to a headed goal from substitute Chris McCann, the first goal that the 18-year-old has scored for the Clarets.
Burnley boss Steve Cotterill was delighted: "That was a big win for us tonight but one we deserved. In the first half they had more of the possession without hurting us, but we gave a big performance.
"Frank Sinclair could have come off after 10 minutes but he stayed on and that typified the performance by everyone. We needed a touch of luck but we will take a deflection as it's the only bit of luck we have had.
"We said to McCann to go on and shoot, we did not realise he would go and head one in."
Ipswich boss Joe Royle stated: "The first goal was vital, a cruel deflection when we were well on top, passing everywhere but not threatening goal.
"As the game wore on they got more and more confident, we gave away a silly free-kick for the second goal and then they were cruising.
"I don't think we would have scored if Burnley came off. We passed the ball but they scored the goals. All match we've woven pretty patterns but they have had the shots."