The striker, who was this week arrested along with his wife as part of a £1.1million tax investigation, was in the right place at the right time for defender George Friend to blast a 90th-minute clearance straight at him and look on as the ball flew into the Boro net.
It was tough luck on the Sky Bet Championship side, who had hit the post before the break through Adama Traore, although the much-changed visitors were significantly better after that to clinch a 1-0 win.
Brighton boss Chris Hughton, perhaps with one eye on Wednesday night's crucial Premier League trip to Southampton, made seven changes to the side which lost 4-0 at home to Chelsea last weekend and saw his team make a tepid start.
Boro striker Ashley Fletcher whistled a second-minute shot a yard wide and as Traore embarked upon a series of threatening forays, the visitors found themselves pinned back deep inside their own half.
However, with striker Sam Baldock providing a focal point and Jiri Skalak prompting from midfield, they gradually worked their way into the game, although keeper Tim Krul was grateful to see Martin Braithwaite head over from Friend's 14th-minute cross as the Teessiders broke at pace.
Fletcher might have done better when Traore picked him out at the near post after powering his way past Markus Suttner and Beram Kayal on the right with 20 minutes played, but the striker was unable to adjust his feet in time and could only send the ball back in the direction from which it had come.
Daniel Ayala should have headed the home side into the lead on the half-hour when he met Braithwaite's corner unopposed, but he directed his downward header straight at Krul.
The defender almost made amends within three minutes when he flicked on Ryan Shotton's long throw to Patrick Bamford, who controlled and then sent an acrobatic overhead kick just too high,
Krul managed to get his fingertips to Traore's thunderous drive five minutes before the break, but still needed the help of the upright after Jonny Howson and Fletcher had combined to allow the midfielder to terrorise Suttner once again, and the half-time whistle came as a relief to Hughton's men.
After a tame start to the second half, it was Brighton who came close to taking the lead with 58 minutes gone.
Randolph spilled Kayal's swerving shot from distance, but redeemed himself by blocking Skalak's follow-up before Ayala threw himself into the path of Baldock's strike after the loose ball fell to him.
Traore had Krul diving full length with a 74th-minute shot which skidded inches wide and Izquierdo fired straight at Randolph on the turn, but the real drama came right at the death.
Friend got to Suttner's cross marginally before Murray, but his attempted clearance cannoned back of the striker and gave Randolph no chance.
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