Sean Dyche's side ended a seven-match winless run thanks to Boyd's second-half strike at Turf Moor and they look on course to preserve their top-flight status.
The Clarets' 10th home league victory this season moved them eight points clear of the bottom three with seven games left.
Stoke's third successive defeat continued their wretched away form, which has seen them lose seven of their last nine on their travels.
At the other end moments later, Mee had a header from Steven Defour's free-kick saved by Lee Grant.
A rapid raid from Stoke's Saido Berahino almost brought the opening goal when he squared to Arnautovic but, with just Tom Heaton to beat, a terrible first touch allowed the Burnley keeper to save easily.
Charlie Adam tested Heaton with a header from Arnautovic's corner just before half-time.
Adam was left red-faced early in the second half when he tripped while taking a corner and picked up the ball thinking he could retake it, instead giving away a free-kick for handball and leaving Hughes shaking his head in disbelief.
There was worse to come for Stoke in the 58th minute as Burnley snatched the lead.
Having laboured for long periods, Burnley finally conjured a moment of class as Jeff Hendrick weaved his way past two Stoke defenders into the penalty area before teeing up Boyd, who kept his composure to score with a clinical finish.
Stoke's Peter Crouch went close with an overhead kick into the side-netting, but the visitors couldn't muster a late equaliser.