Dyche said a possible reason for Burnley's current slump is that they have been slow to get going in matches.
But the hosts were soon ahead, through the Malian worldwide, on 21 minutes.
Zaha then came close to a second when one-on-one with Pope, but his dinked effort was stolen off the toe of McArthur by Ben Mee.
Pope did well to claw the ball from behind him after Benteke had diverted Patrick van Aanholt's drive on target before Sakho's venomous volley, supplied by Timothy Fosu-Mensah, fizzed just wide of the far post.
Ashley Barnes went a touch closer when cushioning Tarkowski's headed assist from Gudmundsson's cross, swinging his left boot at the ball from the edge of the box but Hennessey got a fingertip to the attempt to turn it over.
Sako had retained his place after impressing with a fine goal in Monday's FA Cup third-round defeat at Brighton and quickly demonstrated the confidence with which he is playing with an early run into the Burnley penalty area. In the first half we created enough chances to have made things more comfortable for ourselves in the second half.
Jan Kruger GettyImages
Loan signing Georges-Kevin N'Koudou replaced Jeffrey Hendrick on the hour mark and immediately injected some skill to the Burnley attack, and so almost set up Gudmundsson with an incisive pass, only for McArthur to clear.
Hennessey was out sharply to deny Barnes an equaliser four minutes from time, with a subsequent melee by the touchline after Taylor sent Zaha careering into television equipment indicative of Burnley's frustration.
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.
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