On 19 minutes, the away side did take the lead through Vokes, who poked home from the near post.
© PA Photos
James Henry almost equalised for Millwall, firing over from outside the area after 27 minutes.
Vokes came close to making it 2-0 on 36 minutes, heading just over from Marney's cross.
Burnley doubled their lead after 58 minutes when Danny Ings headed home Kieran Trippier's cross.
Sean Dyche's men almost grabbed a third when Trippier sent a magnificent 60-yard ball into Ings, who just put the ball over with only the goalkeeper to beat.
A minute later, Ings played Keith Treacy in on goal, but he was denied by David Forde in the Millwall goal.