The Clarets finished a dizzying seventh last season, but were desperately out of sorts against a Palace side, who until Saturday's fixture, had failed to score a single goal at home from open play this term.
Unchanged from their goalless draw against Manchester United, Palace were on the front foot from the opening exchanges.More news: Soyuz spacecraft successfully docks with ISS
And Meyer nearly made it two after 35 minutes - his curled left-footed effort beat Hart but went millimetres wide of the post. It came via McArthur, whose cross-shot from the right-hand side crept in off Hart's far post.
With the win, their first at home this season, Palace move up to 15th place and have 12 points.
Burnley's success last season was built on a water-tight defence, but only Fulham have conceded more goals than Dyche's side this term and Meyer almost doubled the home side's advantage before the interval when he was afforded far too much time in the area.
Dyche will have expected a response from his players after the interval, but it never came as a rampant Palace continued to dominate.More news: 1st baby born using uterus transplanted from deceased donor
Wilfried Zaha caused the Clarets defence plenty of problems, forcing two saves out of Hart early in the second half before he crashed a shot against the crossbar.
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.
Burnley looked to have got themselves back on track after a sticky start with back-to-back wins over Bournemouth and Cardiff but they have lost four of their last five games, with Monday's setback against Newcastle the latest disappointment.
"We have had chances at home, and created more chances than we have away".More news: Tom Cruise Is Going To War With A Setting On Your TV
The second goal finally arrived as Townsend smashed home a attractive left foot shot to put Palace 2-0 up with their 27th attempt of the game.
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