Neither side took a stranglehold on the game in the early stages, with both having spatters of attacks but nobody causing much of a threat in the final third.
Arnautovic squandered a big opportunity to nudge the Potters ahead when Saido Berahino played him through on goal, but the Austrian's first touch was too heavy, which allowed Tom Heaton to race off his line and scoop up the ball.
The visitors looked threatening on the break, and they almost scored from one move after Arnautovic's shot got deflected for a corner.
As the ball was whipped in from the flag, Charlie Adam connected at the near post, but his header was stopped by Heaton, who was the busier of the two keepers in the first half.
Adam tried another approach as he charged forward before unleashing a 35-yard powerful shot, but he did little to trouble the Burnley keeper as the ball was hit directly at him.
After the break, Stoke jumped out of the blocks and could have gone ahead through Adam, who scuffed two attempts - one from on the edge of the penalty area and one from inside.
Arnautovic also came close when Mee's failed clearance allowed the Austrian to get a header away, but he executed it poorly as the ball fell wide.
The breakthrough was made in the 58th minute when Jeff Hendrick's individual run down the left flank ended with him sending a low cross across goal, where Boyd was placed to slot home.
Burnley were lifted by the goal and Hendrick almost nabbed one of his own when his first-time volley was directed towards the bottom corner.
The ball was deflected off Ryan Shawcross, but as a result, the sting was taken out of the pace and Grant was able to scramble across his goal to gather.
In the closing stages of the encounter, Burnley camped themselves in their own half in a bid to hold on to their lead, and they were able to do so as Stoke failed to nab an equaliser.
The Potters have now suffered three defeats on the trot, and as a result drop to 11th in the table, while Burnley move up to 12th with seven games left to play.