Goals from Ashley Westwood and Chris Wood in the first half carried the Clarets to maximum points, which pushed them 12 points clear of the bottom three with just three games remaining.
Sheffield United are the other team who will be demoted this term, with the Blades having long been cut adrift at the foot of the table.
At the opposite end of the standings, meanwhile, Manchester City are still waiting on confirmation that they will finish the season as champions, though that might arrive on Tuesday when Manchester United play Leicester. Anything less than a victory for the Red Devils will see their neighbours seal a third title in four years.
The race for the top four, meanwhile, is not over. Much rests on the Foxes, however. Should they win at Old Trafford, it is unlikely that either them of Chelsea will be caught.
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