Without a win in six home games heading into Saturday's encounter at Selhurst Park, Palace took a deserved lead when McArthur's cross found its way into the net 16 minutes in.
Wilfried Zaha and Luka Milivojevic squandered gilt-edged chances to extend Palace's lead after the interval, while Charlie Taylor's wonderful block denied McArthur a second.
Zaha's effort against the crossbar kept Burnley in with a chance, but any lingering hopes of a comeback were extinguished when Townsend hammered home with 13 minutes remaining.
Cheikhou Kouyate and Max Meyer went close as Palace started strongly, though it would take a stroke of luck to give the hosts the lead.
McArthur's inswinging cross from the right evaded Meyer's touch, but Joe Hart's hesitation enabled the ball to skid across the face of goal before clipping in off the upright.
Having forced a fine save out of Hart prior to the interval, Zaha should have done better when played through one-on-one with Burnley's goalkeeper, only to blaze over from 18 yards.
Milivojevic failed to keep a close-range effort on target as Palace continued to press, before McArthur had a goal-bound attempt blocked by Taylor.
The woodwork came to Burnley's rescue when Zaha unleashed a dipping shot from the edge of the box, but Townsend had no such bad luck when he curled an exquisite effort into the top-left corner to make sure of the points from a one-sided affair.
What does it mean? Palace move clear
Roy Hodgson's side are now on 12 points, three clear of the relegation zone, and the Eagles will be confident heading into three winnable fixtures before they face Manchester City on December 22. Burnley, meanwhile, remain in 18th place, a point behind Huddersfield Town.
Meyer comes to the fore
It has been a stuttering introduction to life in England for Meyer, but the former Schalke playmaker demonstrated his quality with a slick performance on his fifth Premier League start, and it was his intelligent run that caused havoc for Burnley's defence in the lead-up to McArthur's opener.
Defensive woes have Clarets reeling
Burnley's defensive resilience was crucial to their success last term, but Dyche's side have now conceded 29 goals, with only Fulham having let in more.
Key Opta Facts
- Crystal Palace mustered 29 shots in the game, more than they have managed in any Premier League game since 2003/04.
- Burnley have enjoyed just two wins in their last 19 league games (D5 L12), after winning five in a row before that run.
- Palace registered their first victory in their last nine league games (D3 L5).
- The Clarets have not won any of their last 22 away top-flight matches in December (D7 L15) since a 1-0 win in December 1969 against West Brom.
- Townsend has scored and assisted in the same Premier League game for only the fourth time and the first since December 2017 (for Palace against Arsenal).