The Whites boosted themselves eight points clear of the Clarets and top flight drop zone at Elland Road on Sunday.
Half-time substitute and key visiting man Maxwel Cornet levelled in the 54th minute with a stunning free-kick from distance.
United rallied, though, and it was Dallas who mustered a wonderful goal from outside the box following a short corner routine from Raphinha and Mateusz Klich.
The Northern Irishman’s effort sent the home crowd wild 13 minutes from time before Dan James sealed the spoils with a header late on.
“We knew coming into this game it was huge – people talk about six-pointers, but we’re coming off the back of a few heavy defeats, so we’re delighted,” Dallas reflected post-match.
“I thought we dominated. We conceded – it was a great strike from the free-kick, but we got back into it after that. It’s a relief more than anything, we didn’t start so well, they caused us some problems. We got into the game then and we dominated.
“We’ve tried to rediscover some of the form we’ve been missing this season. People talk about Burnley being a long-ball team, but there’s a method to the way they play, they’re difficult opposition.”
Asked about his strike which proved to be a vital moment in the game, he said: “I saw a wall of players coming towards me, so I tried to just bend it in – some go for you, sometimes they don’t. I hadn’t that much time to think about it and thankfully it came off.
“It’s massive, getting goals, we’ve been missing our main striker [Patrick Bamford] quite a bit. We’ve had a lot of issues throughout the festive period, we don’t use it as an excuse, but credit to the staff here for getting players out there to play today.”