After Dwight McNeil's early shot had been saved by Karl Darlow, Newcastle led on 14 minutes through a great run and finish by Saint-Maximin.
Sean Dyche's team stepped up their search for an equaliser after half-time and it deservedly came when Ashley Westwood volleyed in on 61 minutes.
Callum Wilson restored Newcastle's advantage just four minutes later, converting a cross from Saint-Maximin, who limped off shortly after.
Wilson made it 3-1 with a penalty on 77 minutes after Nick Pope had fouled Ryan Fraser to move Newcastle up to sixth on seven points.
Burnley are 19th without a point.
Newcastle: 17 Oct v Man Utd (H)
Burnley: 17 Oct v West Brom (A)