The centre-half was forced to take the gloves after David James suffered an injury on the hour mark, with Robins boss Derek McInnes opting not to name a goalkeeper among his substitutes.
It looked as if they would be punished as 13 minutes later Fontaine was picking the ball out of the net after being beaten by a superb 30-yard strike from Danny Ings, but the visitors rescued a surprise point and it was a great moment for striker Ryan Taylor, who celebrated his first appearance since mid-October by levelling from 25 yards for his first goal for the club.
The result left the Clarets in 13th, while City ended the season in 20th position.
The Robins had secured their safety last weekend and so with nothing to play for, the shackles were off and it was an open start to the game.
The first chance fell to the visitors, who should have taken the lead in the 12th minute when Brett Pitman found himself in a good position from Martyn Woolford's cross, but he headed straight at Brian Jensen in the Burnley goal.
Burnley responded and Ross Wallace fired over from a tight angle after good work from Ings before City were back on the attack and home debutant Danny Lafferty made a superb tackle to deny Chris Wood.
It was more of the same for the remainder of the half with both sides pushing, but it was the Clarets that had the best chance to open the scoring on the brink of half-time. Lafferty found Charlie Austin and after he beat the last man, he bore down on goal, but did not even test James in the Bristol City goal, putting his shot wide.
Burnley continued in the ascendancy after the break and created a number of chances in the opening 15 minutes of the second period.
Austin had a goalbound effort blocked by Fontaine, Marvin Bartley tested James with a drive and Wallace was again off-target.
James was not involved in the action for much longer, though, as he suffered an injury and with no reserve goalkeeper on the bench, defender Fontaine was forced to go in goal for the final 28 minutes.
Burnley sensed a season-ending win was on the cards and after Austin had another effort blocked, they finally took the lead and did so in fine style.
Lafferty was again involved, finding Ings and he smashed a 30-yard effort sailing past a helpless Fontaine with 15 minutes to go.
The lead did not last long, though, as Taylor curled in a classy effort from 25 yards to celebrate his first goal for the Robins since his summer move from Rotherham and that is how it ended as Fontaine avoided any further scares.