The West Countrymen were two points in arrears going into the latter stages, but the hosts appeared set to win when Burns strolled over the line, only for the full-back to have the ball knocked out of his hand when already celebrating.
Todd Blackadder’s men had enjoyed a positive first half-hour, going 17-7 in front via Jamie Roberts and Burns tries, but Les Rouge et Noir kept themselves in the contest at the break.
Sofiane Guitoune crossed the whitewash, to go with Maxime Medard’s earlier effort, as they gave themselves hope heading into the final 40 minutes.
Burns and Thomas Ramos traded penalties at the start of the second period, but Guitoune took Toulouse in front with 18 minutes remaining.
Bath then had several opportunities to win the game but Burns missed a late kick – his third of the contest – before the full-back’s embarrassing moment cost his side at the end.
The West Countrymen have struggled to make an impact in Europe’s top-tier competition over recent times and they will rue this missed opportunity.
They opened the scoring through Burns but Les Rouge et Noir hit back after the full-back was turned over at the breakdown. There was plenty of space on the left and the visitors eventually took advantage when good hands sent Medard across the whitewash.
Unperturbed, the hosts upped the intensity and started to go slightly more direct. Dave Attwood provided the thrust and, with the opposition defence in reverse, Roberts took a neat line to score.
Bath were now very much in the ascendency and touched down for the second time when Jackson Willison and Henry Thomas combined. The ball was shifted wide and Burns brilliantly shrugged off the attentions of two tacklers to finish.
Cheslin Kolbe was one of those who failed to stop the full-back, but the recently capped South African international gained his revenge by creating Guitoune’s try after stepping inside two defenders and passing to the centre.
That meant Toulouse went into the interval just five points in arrears but they were dealt a blow following Jerome Kaino’s yellow card for a high tackle.
Ugo Mola’s men handled the sin-bin period well, however, despite the West Countrymen managing to extend their advantage through another Burns three-pointer.
They were duly rewarded for their efforts during Kaino’s time off the field when Tom Ellis was also yellow carded for dangerously sending Rynhardt Elstadt beyond the horizontal.
Unlike Bath, who failed to benefit from the opposition being down to 14, the visitors scored 10 points while Ellis was absent. Ramos initially reduced the arrears from the tee before the French outfit moved ahead through Guitoune’s well-taken try.
Confidence was high in the Toulouse ranks but they bizarrely looked to move the ball from deep and it invited pressure.
It meant Blackadder’s charges could go in search of the winning points and the hosts thought they had it, but Burns’ mistakes meant they had to settle for just a losing bonus-point.
Tries: Roberts, Burns
Cons: Burns 2
Pens: Burns 2
Yellow Card: Ellis
Tries: Medard, Guitoune 2
Cons: Ramos 2
Yellow Card: Kaino
Bath: 15 Freddie Burns, 14 Semesa Rokoduguni, 13 Jackson Willison, 12 Jamie Roberts, 11 Darren Atkins, 10 Rhys Priestland (c), 9 Chris Cook, 8 Zach Mercer, 7 Francois Louw, 6 Tom Ellis, 5 Luke Charteris, 4 Dave Attwood, 3 Henry Thomas, 2 Tom Dunn, 1 Nathan Catt
Replacements: 16 Jack Walker, 17 Jacques van Rooyen, 18 Anthony Perenise, 19 Elliott Stooke, 20 Sam Underhill, 21 Max Green, 22 Max Wright, 23 Tom Homer
Toulouse: 15 Thomas Ramos, 14 Cheslin Kolbe, 13 Sofiane Guitoune, 12 Maxime Mermoz, 11 Maxime Médard, 10 Zack Holmes, 9 Sébastien Bézy, 8 Jerome Kaino, 7 Rynhardt Elstadt, 6 Francois Cros (c), 5 Joe Tekori, 4 Florian Verhaeghe, 3 Charlie Faumuina, 2 Leonardo Ghiraldini, 1 Lucas Pointud
Replacements: 16 Julien Marchand, 17 Clément Castets, 18 Maks Van Dyk, 19 Pierre Gayraud, 20 Selevasio Tolofua, 21 Louis Madaule, 22 Antoine Dupont, 23 Romain Ntamack
Referee: Andrew Brace (Ireland)
Assistant referees: John Lacey (Ireland), Eddie Hogan-O’Connell (Ireland)
TMO: Simon McDowell (Ireland)