The Clarets keeper somehow managed to keep out the Spaniard's 67th-minute effort to preserve his clean sheet and assure his side of just a fourth league point on the road this season.
However, Heaton's heroics left Boro, who are now without a league win in 14 attempts dating back to December 17, in ever deeper trouble with just seven games remaining.
Having emerged from clashes with relegation rivals Swansea and Hull in fixtures which had been ear-marked as life-savers with only a point to show for their efforts, Steve Agnew's men knew they were drinking in the last chance saloon.
Accordingly, the interim head coach tinkered once again with both personnel and system.
Wing-backs Stewart Downing and Antonio Barragan exploited the space they were afforded well to produce a series of crosses, but the Teessiders were unable to muster the quality they needed to make the most of their possession when it mattered most with Heaton largely untroubled before the break.
He saw full-back Stephen Ward block a sixth-minute Barragan shot at source and striker Rudy Gestede glance a 28th-minute header well wide, although he heaved a huge sigh of relief four minutes before the break when defender Daniel Ayala blasted the ball across his six-yard area after he had recycled Grant Leadbitter's free kick, with no team-mate able to make meaningful contact.
The Clarets saw less of the ball, but with Joey Barton composed in the middle of the field, used their share to better effect with the impressive Robbie Brady clipping the top of Victor Valdes' crossbar with a 14th-minute free-kick and George Boyd blasting wastefully over on the turn 12 minutes later.
Gestede forced Heaton into his first real save of the game with a downward header from a 47th-minute corner and Downing sent a volley wide after Leadbitter's corner had been cleared four minutes later, but the visitors should have been in front with 57 minutes gone.
Ward evaded Barragan's sliding challenge to pull the ball back and when Ashley Barnes stepped over the ball, Brady seemed certain to score, but dragged his left-foot effort wide.
Valdes had to pluck Ben Mee's goal-bound header from underneath his bar six minutes later with Burnley pressing for the breakthrough, but it took a magnificent reaction save from opposite number Heaton to keep out substitute Negredo's stinging volley from Downing's 67th-minute cross.
The Burnley keeper was on hand to field Adam Clayton's weak shot following substitute Adama Traore's mazy run and pull-back five minutes later, but there was further drama to come.
Referee Martin Atkinson, who had earlier booked Stephen Ward for an off-the-ball clash with Barragan, produced a yellow card for last man Michael Keane's foul on substitute Patrick Bamford with the home fans baying for a red.
But Matthew Lowton headed Downing's resulting free-kick off the line and Negredo saw a late strike chalked off for handball as the visitors clung on to their point.