Jake White's charges stood up well to Leinster's initial attacks, but Sheehan turned a Ross Byrne kick through into an opportunist ninth-minute try.
The Bulls, who got on the board with a Chris Smith penalty, ramped up their physicality to touch down in the 19th and 24th minutes.
Canan Moodie's knock-on over the try-line was forgiven when hooker Grobbelaar barged over from a tap penalty. Smith converted with the aid of the post.
Elrigh Louw's powerful carrying soon got them back into scoring range, Coetzee the beneficiary of Zak Burger's short pass to go in under the posts to make it 17-7.
Leinster's backs brilliantly got on the outside in the 32nd minute before Henshaw stretched out to score. Byrne's conversion restored the three-point gap.
Number eight Louw again led the charge for the Pretoria-based side on the restart. Grobbelaar controlled an advancing maul and Leinster coughed up a penalty try with Ryan binned for the collapse.
Not even the springing of Johnny Sexton could inspire Leinster. A sharp attack gave them a lifeline with 11 minutes remaining as O'Loughlin was released for the left corner.
However, replacement Morne Steyn's 76th-minute penalty had the Bulls out of reach before Healy burrowed over past the 80-minute mark.
Player of the match
The Bulls came with a smart game-plan, shutting down Leinster's attack as much as they could and rattling the province's lineout, in particular.
A number of their players excelled in their 10th win in 11 URC games, none more so than Johan Grobbelaar who took his try well, almost had a second and was at the heart of a strong night for their set-piece.
There was much to admire about the Bulls' directness, with their power-packed forward play rewarded with three tries.
However, Leinster's fightback produced a cracking team score. Sharp passing released Jordan Larmour and Garry Ringrose into space, and Jimmy O'Brien fed O'Loughlin to finish off in the left corner.
