The home side made the most of their next opportunity, however, as they went ahead in the 20th minute.
Burnley left-back Charlie Taylor cleared a viciously dipping Patrick van Aanholt cross just over his own crossbar in the eighth minute - a close shave that encapsulated a nervy opening for the visitors.
From a similar position in the 21st minute he gave Palace the lead. Sako cut in from the left hand side and arrowed a shot through the legs of Tarkowski and past a stranded Nick Pope. Captain Ben Mee flung himself bravely to brilliantly deny James McArthur on the goal-line.
Palace continued to dominate their opponents as Benteke nodded over from a Luka Milivojevic corner, while Sako volleyed just wide from a deep cross from Timothy Fosu-Mensah.
The latest success was built on Bakary Sako's well-taken first-half goal, some sparkling moves before the break and a more resilient performance in the second half, when Burnley stirred and began to pose a threat.
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Georges-Kevin Nkoudou was introduced in place of Jeff Hendrick for a Burnley debut after the hour and the Tottenham loanee nearly had an immediate impact, clipping in a cross from the byline on the left that Gudmundsson was unable to turn home.
Hennessey then had to be on his toes to rush out of goal and block Barnes as Burnley edged towards an equaliser.
Arfield is a very good midfielder who has proved himself in the Championship and in the Premier League in England.
Andros Townsend and Jeffrey Schlupp have been added to Crystal Palace's lengthy injury list and will miss this weekend's Premier League fixture against Burnley, according to the club's official website.
West Ham and Crystal Palace are said to be interested in his signature, though the Daily Record are reporting that Rangers are also monitoring the situation in the hope that Arfield would prefer a return to his native Scotland.