Louis van Gaal handed Angel di Maria a first start since his arrival from Real Madrid.
It was Jones who almost gave the Clarets a dream start just five minutes in as he curled a superb free-kick around United's wall, only to see the effort hit David de Gea's crossbar.
The midfielder was at it again midway through the first-half, capitalising on some sloppy United defending to volley a stinging drive which forced De Gea into a superb save.
Between those efforts, Van Persie produced a smart save from Heaton after collecting a pass from the promising Di Maria.
Burnley finished the stronger of the two teams, with the home fans enjoying a chorus of 'ole's' towards the end of the first-half as they dominated possession.
Neither manager chose to make a change at the break and it was United which began the second-half the sharper.
With Di Maria's pace testing the impressive and combative Michael Duffy, it was the Argentine who carved out a wonderful chance for Van Persie, barreling his way down the wing before his square ball eventually fell for the Dutchman.
Van Persie controlled his volley cleverly, but saw the effort cleared off the line by Dean Marney.
Di Maria enjoyed a physically testing clash with Duffy and eventually was forced to make way for Anderson on 70 minutes complaining of cramp.
Five minutes later Van Persie was also off for Danny Welbeck.
On 80 minutes, Rooney missed the best chance of the game, finding himself clear at an Anderson corner, only to head wide of Heaton's goal.
Mata handed Van Gaal further injury concern minutes later, complaining of a hamstring strain. He was replaced by Adnan Januzaj.
Inside the final five minutes, United felt they deserved a penalty after substitute Ashley Barnes appeared to handle inside the Burnley penalty area.
With United beginning to push for a winner, Dyche threw on Steven Reid for his Clarets debut and the veteran 'took one for the team' as he was booked for denying Welbeck a clear run after breaking from a Burnley corner.
The match ended with United pushing for a winner, but Burnley's defence held firm to get their season underway.
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