The centre-back has signed a four year deal at Turf Moor and joins for an undisclosed fee, reported to be in the region of £2 million.
The Irishman joined The U's from St Patrick's Athletic back in January 2021 and enjoyed a breakthrough season in League One, making 30 appearances for Karl Robinson's side.
McNally, 22, had youth spells at both Enfield Athletic and Drogheda United, with both clubs being local to the defender.
After two seasons at Drogheda, the centre-half moved on to St Pat's, where he signed his first professional contract.
Following this, he rejoined Drogheda United on loan, in a season where he earned their Young Player of the Year award, as well as scoring a perfect hat-trick in The Drogs in their promotion play-off semi-final.
An impressive feat for a centre-back for sure.
On his return to St Pat's, McNally won another Young Player of the Year award, before an inevitable move across the border materialised, as he joined Oxford United.
What will he bring to the Clarets?
McNally, in some ways can be compared to current Burnley defender and fellow Irishman Nathan Collins.
The ex-Drogheda United man likes to dribble with the ball and drive out of defence, and looks to be successful in doing so a lot of the time.
McNally's 6ft 4in height can also make him key in the air for Vincent Kompany's side, both at the back and going forward from set-piece situations, and with new man Scott Twine in the side, expect Burnley to succeed from those.
With the potential departure of Collins himself, coupled with the potential incomings of fellow defenders Taylor Harwood-Bellis and CJ Egan-Riley from Manchester City, it is unknown what sort of role McNally may play.
However, with boss Kompany already stating his desire for permanent transfers rather than loan deals, as Harwood-Bellis is expected to be, McNally could be held in higher stead than these arrivals.
The step-up from the third to second tier is a big one, similar to fellow new signing Twine, however with a full pre-season under legendary centre-back Kompany's tutelage, the 22-year-old should be able to adapt with ease.
A ball-playing defender in McNally should suit the new Burnley 'style' and could be key in The Clarets push for an immediate return to the top-flight.
What the club and McNally have said about the move
Kompany was happy to add yet another new signing to his side before the beginning of the new season.
"Luke is a good player, he's good on the ball, wins his challenges, and knows how to defend," Kompany told the club's official website.
"We are pleased to welcome him to our Club.
"He has shown all the qualities we were looking for in a central defender, and we are looking forward to working with Luke."
McNally was equally as happy to make the move, which gives him the potential to make a second-tier debut against Huddersfield Town on July 29.
"It's unreal to be here, I'm absolutely buzzing and so glad to have got the move done," said McNally, adding to his new managers comments.
"It's all happened so quick really, I just can't wait to start now and I'm really excited."
Burnley will hope to add further to their squad in the coming weeks, but the 'new era' is truely up and running.
McNally has signed a four-year-deal and has joined up with the squad ahead of their opening pre-season game against Rochdale, which is behind closed-doors on Friday, July 1.