The future of Murray following Valerien Ismael's appointment as head coach on Friday remains to be seen but he will have been pleased with what was a battling performance from the Tykes at The Den.
The Tykes can be forgiven for being slow to start after that disruption, but Millwall themselves were under a malaise during an opening quarter that saw neither team have so much as a half-chance.
Barnsley goalkeeper Jack Walton almost injected some excitement into what had been a dire first period when he came racing out of his area but, fortunately for him, he just about beat Jed Wallace to a loose ball.
Completely out of the blue, Mowatt lashed the visitors in front after 44 minutes when he arrowed a stunning shot into the top corner from 25 yards after a cross was headed out into his path.
The Tykes couldn't hold their lead until half-time, however, as Connor Mahoney's corner found Cooper at the back post and Walton was unable to keep out his header.
Walton had to have his wits about him, though, as he had to be on hand to turn Jed Wallace's free-kick from out on the left over the bar in the early stages of the second half.
The Lions were then frustrated by the woodwork for a second time when Scott Malone's looping header from Bennett's cross came back off the bar.
Walton then came to Barnsley's rescue again in the 90th minute when he produced a superb save to keep out Kenneth Zohore's header.