A brace from Vokes and a well-placed effort from Stephen Ward proved enough for Burnley collect all three points from Stamford Bridge despite Chelsea hitting back in the second period through Alvaro Morata and David Luiz.
Vokes has claimed that the Clarets "were sloppy at times", but the Wales international has branded the win "a huge result" as Burnley look to fight off the threat of relegation once again this season.
"We were sloppy at times but we ground the result out and we're delighted as a result," Vokes told reporters. "That's a huge result - a lot was made of our away form last season. So that result is huge, a massive win for us at the start of the season. It was always going to be tough here but we got our heads down and put in a performance."
Vokes, 27, scored 10 Premier League goals for Burnley during the 2016-17 season.