Summer signing Jutkiewicz continued his fine form in pre-season as he made it five goals in five games since his move from Middlesbrough.
The home side hit the front after just two minutes when Jutkiewicz collected a pass from Ross Wallace and fired home past Ruben Blanco.
After several changes from both sides in the second half, Celta drew level on 69 minutes with Alex Lopez finding the net for the visitors.
However, parity proved short-lived as Burnley regained the lead just 60 seconds later as Ashley Barnes set up Jutkiewicz for his second goal of the night.
Celta quickly drew level again as Nolito replied for Celta and ensured the game ended all-square.
Star striker Danny Ings missed the game through injury, but Sean Dyche played down the extent of the problem insisting he did not want to take any risks with the player ahead of the Premier League’s big kick-off next week.