David Buchanan scored just before the break to give bottom side Accies the interval lead but in a remarkable 10-minute turnaround at New St Mirren Park, Higdon scored with a header, a penalty and a wonderful volley.
Hamilton, who had defender Martin Canning sent off when he conceded the spot-kick with a challenge on Buddies' substitute Craig Dargo, are nine points behind the 11th placed Paisley side and have played a game more.
With games running out, and on the evidence of their second-half collapse, Billy Reid's side have to hope for a football miracle if they are to beat the drop.
The early stages showed why both clubs were at the wrong end of the SPL table.
In a frantic start to proceedings neither side could get any sort of passing game going although admittedly they were not helped by the blustery conditions.
The first moment of concern for Accies came in the 13th minute and was self-inflicted.
Canning's pass back to Tomas Cerny was too high and the Hamilton keeper had to head clear to avoid the ball going behind for a corner.
Accies forced their first corner in the 17th minute through the efforts of the lanky Tom Elliott, who was leading the line with Nigel Hasselbaink, but it came to nothing.
The ball was in the air far too often for any sort of pattern to develop but in the 26th minute it was the visitors who created the first clear-cut chance of the game.
Hasselbaink drove down the right and his clever through ball found Jim McAlister racing into the St Mirren box but his close-range shot was parried high into the air by Buddies' keeper Paul Gallacher who gathered at the second attempt.
St Mirren fans' simmering discontentment became audible when
Higdon sent a drive from distance past Cerny's right-hand post five minutes later, when he had better options.
Five minutes from the break Accies midfielder Flavio Paixao broke past a weak tackle by Saints defender Darren McGregor to go through on goal alone but his attempted chip was beaten away by Gallacher and Ali Crawford drove wide of the target from the edge of the box.
The immediate thought was that it might prove to be a costly miss but the small band of travelling fans were in raptures just before the break when Buchanan scored from 20 yards after Hasselbaink's shot had been blocked.
The Lanarkshire side bounded out the tunnel for the second half and within four minutes Gallacher had to make a decent save from Elliott's header following an in-swinging cross from McAlister.
St Mirren increased their efforts in a bid for the equaliser.
In the 52nd minute Kenny McLean's corner from the right was headed back across goal by McGregor and at the second attempt Buddies' defender sent a toe-poke towards goal which Cerny saved.
Moments later, Jure Travner's cross from the left was missed by Paul McGowan but it still fell to Steven Thomson at the back post but the midfielder's right-footed drive hit the side-netting.
In the 59th minute Dargo replaced former Hamilton midfielder Jim Goodwin as the Buddies tried a more positive approach and two minutes later it paid off when David van Zanten's cross from the right was met by Higdon who dived to head past Cerny from six yards.
In the 63rd minute there was more trouble for bewildered Accies when Canning was red carded by referee Willie Collum for a foul on Dargo as the former Kilmarnock striker broke in to the Hamilton box, and Higdon swept in the spot kick.
In the 71st minute Higdon completed his hat-trick in memorable style when he volleyed into the top corner from 14 yards after racing on to Dargo's raking pass which had evaded Accies backtracking skipper Alex Neil.
Hamilton looked shattered and Dargo flashed a drive over the bar just before Higdon was given a standing ovation when he was replaced by Patrick Cregg.
St Mirren were mostly content to keep possession in the final minutes happy in the knowledge that their SPL status was probably secure for another season.