Olam kept the Tigers in Saturday's game with a try-scoring double, before youngster Fonua Pole sealed victory for Benji Marshall's men with a 76th-minute winner.
The nail-biting victory in front of 8392 fans at Leichhardt Oval marked the first time since April 1 that the Tigers had won an NRL match.
And the result lifts the Tigers off bottom place, with Gold Coast now occupying last spot.
Captain Api Koroisau, playing out of position at halfback, was a standout for the Tigers in a team that boasted two debutants and three players with fewer than 10 NRL games.
But Olam, who has been battling a knee injury all season, stood tall with two tries after the Titans eased to a fifth-minute lead through Jojo Fifita.
The PNG international finished off a well-worked backline move by weaving past a handful of Titans defenders to send the crowd on the hill into a frenzy.
But there was a collective groan from the legion of Tigers supporters when Kieran Foran drifted through a Pole tackle to set up an on-rushing Jayden Campbell.
Olam pulled off a last-ditch tackle on Titans livewire Keano Kini to restrict Gold Coast's halftime lead to 10-6, before clattering the Titans fullback in trademark fashion early in the second half.
Neither side was able to take control of the game, but the breakthrough came when Tigers back-rower Samuela Fainu trampled over Foran and popped a one-handed offload away for Olam to finish in the corner.
Koroisau kicked the touchline conversion to give the home side a two-point lead with 20 minutes left.
In a dramatic final quarter, Olam was held up just short in the pursuit of his hat-trick, and Fifita fumbled the ball two metres out from the Tigers' line.
Fifita's blunder would prove costly as Stefano Utoikamanu bulldozed his way up-field to lay the platform for the winning try.
Koroisau trickled a grubber-kick towards the in-goal, with Pole picking up the ball and touching down to give the Tigers the win.
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