The Dragons found some success with their call to drop Lachie Ilias for rookie Lykhan King-Togia but it was rival halfback Jarome Luai who had his fingerprints all over a match-defining first half on Saturday night.
Already down 10 points, Saints had their spirits broken on the stroke of half-time after Jaydn Su'A dived over the line to score what would have been the Dragons' third try.
But just before Su'A hit the turf, the ball inexplicably dropped from the second-rower's grasp without a defender touching him in the wet conditions.
Bula collected the footy and sprinted from the in-goal to score under the posts just before the siren sounded.
In a flash, the Dragons went from what could have been four points down had Su'A scored, to 16 points in arrears and hanging by a thread.
The bizarre turn of events proved pivotal to the end result when the Dragons fought back into the game in the second half.
Bench hooker Jacob Liddle crossed for two tries and closed the deficit to one score, continuing his excellent season to date.
Bula's miracle try also came after he was left struggling to run when his ankle twisted awkwardly in a tackle from Clint Gutherson and Val Holmes.
The sides traded four-pointers in the final 10 minutes as Adam Doueihi and then Dragons forward Hamish Stewart crashed over but the Tigers never gave up their lead.
Replacing the axed Ilias at halfback, teenager King-Togia looked the goods early in his fifth NRL game, sending a flat pass to put Dylan Egan over for the game's first points.
He blew the Dragons' next set in the red zone sending a no-look flick pass straight to Tigers winger Charlie Staines but put Liddle in for his second try after Saints made a break down the left side.
King-Togia now has the same number of try assists for the season as Ilias managed in seven games, according to the NRL website.
Four-time premiership-winner Luai continued to push his NSW State of Origin case laying on two tries in the space of three minutes in the first half.
He put a kick on to the left side that Doueihi fielded uncontested, passing the ball left for Staines, who confirmed a tryscoring double.
After a penalty in the set after points, the Tigers marched up the other end where Luai threaded a grubber kick past the line for Samuela Fainu to score.
The result means that if highly-fancied Canterbury beat Gold Coast on Sunday, the Dragons will finish the weekend as the only side never to have won at Magic Round.
Tigers centre Brent Naden left the contest early in the second half with a knee injury, with Dragons forward Emre Guler joining him on the sidelines with a shoulder injury later on.
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