Burnley had led through an 18th-minute goal from striker Matej Vydra and were well on top before Saint-Maximin, returning from injury, came off the bench and changed the game inside seven minutes with an assist and a goal.
The result will ease the nerves of Newcastle fans whose team are now six points above Fulham, who occupy the final relegation spot. Steve Bruce's side, in 17th place, also have a game in hand and are just a point behind 15th-placed Burnley.
Sean Dyche's team still have work to do to make sure of top- flight survival and will be bitterly disappointed to have got nothing from a game they dominated for large stretches.
The Clarets made a lively start and broke through when Chris Wood did well down the left and pulled the ball back to Vydra who confidently slotted home.
Dwight Gayle had the ball in the net for Newcastle but was flagged offside and Burnley keeper Bailey Peacock-Farrell, in for the injured Nick Pope, did well to keep out a near-post flick from the forward.
In an end to end game, Newcastle keeper Martin Dubravka did well with a reflex save at close range to keep out a potential own goal from Paul Dummett and he was alert to tip over a thundering effort from Matt Lowton.
Burnley put Dubravka's goal under intense pressure and needing a change of momentum Newcastle manager Steve Bruce made a double substitution in the 57th minute, sending on Callum Wilson and Saint-Maximin who were both returning from injury.
The move worked in spectacular fashion as after being on the field for two minutes Saint-Maximin set up Jacob Murphy for a superbly struck equaliser from outside the box.
Five minutes later, the Frenchman showed his individual ability with a brilliant goal to put Newcastle ahead, picking the ball up on half-way and running at the Burnley defence before jinking to create space and unleashing a lethal drive into the bottom corner.