Prior to limping off with a first-half injury, Bryan Mbeumo had given Thomas Frank's men the lead from the spot, but Pascal Gross quickly levelled matters before 18-year-old Hinshelwood deservedly got the well-drilled Seagulls back to winning ways in the top flight.
Goalmouth action was at a premium between two well-matched teams in the early stages, and it took 18 minutes for a meaningful effort to be registered, but Joao Pedro's looping header was meat and drink for Mark Flekken.
The hosts were beginning to turn the screw, but an intricate piece of Bees play in the 25th minute ended with referee Peter Bankes pointing to the spot, as Vitaly Janelt sauntered into the penalty area and was felled by the arm of Jan Paul van Hecke.
The Brighton defender pleaded his innocence, but Bankes's decision stood, and Mbeumo took a casual run-up - with a Jorginho-esque hop - before nonchalantly rolling the ball into the bottom right-hand corner as Jason Steele dive the opposite way.
However, Brentford's advantage was wiped out in a matter of moments, as with 31 minutes on the clock, Kaoru Mitoma teed up Gross just outside the box, and the German took one touch before picking out the far corner with a crisp left-footed strike; Flekken could only stand and watch as Gross's effort sailed past him.
Injury was added to insult for Brentford just seven minutes later, as Mbeumo appeared to twist his ankle and had to be helped off the field to be replaced by Yoane Wissa, adding to the Bees' already alarming injury crisis.
Van Hecke attempted to make amends for giving away the penalty for Mbeumo to convert, but the Brighton defender sent a header from a corner horribly wide in the 41st minute, and an injury-time Mitoma strike also sailed over the top before the half-time whistle went.
Picking up where they left off, Brighton celebrated an unlikely hero in the 52nd minute, as mere seconds after clearing a Wissa shot off the line, Hinshelwood rose highest to nod Gross's cross across goal and into the far corner for his first senior goal.
The Seagulls remained in the ascendancy as they sought to increase their cushion, and Billy Gilmour was incensed in the 76th minute, as Wissa got away with a yellow card for putting his hand in the Scotsman's face - replays showed some force was used by the Brentford winger as he took a swipe at the midfielder.
Brighton were happy to retain possession in the closing stages, but Flekken had to get down low to save a Simon Adingra volley in the 81st minute, and the Brighton attacker's bad luck in front of goal continued two moments later, as his curling strike from the edge of the box went an inch wide of the post.
A speculative Ben Mee overhead kick flew wide of the mark as Roberto De Zerbi's side remained eighth in the table, while Brentford stay 11th but now have just a one-point buffer over Wolverhampton Wanderers and Fulham.
Brighton now prepare to welcome Burnley to the Amex Stadium on Saturday, while Brentford have a trip to Sheffield United pencilled in for that afternoon.No Data Analysis info