St George Illawarra led by as much as 22-6 late in the first half, but didn't pull away until late and were ahead by just four points when Molo was sent from the field for a high tackle on Patrick Herbert, which ruled the Titan out for the remainder of the game.
The Gold Coast meanwhile failed to take advantage of their numerical advantage in the closing exchanges, and in the end collapsed badly to leak three tries in that period.
In addition to Moga's three, Jack Bird and Talatau Amone crossed for doubles, taking the latter's personal tally to five in two games, following his own hat-trick a week earlier.
The Titans led the match just once after AJ Brimson's try was converted by Tanah Boyd, following Amone scoring the opener, before Moga crossed to put the Dragons back in front.
He was a blade of grass away from having a second three minutes later too, but the NRL Bunker found his knee had brushed the ground over the sideline just before he touched the ball down.
But the Dragons didn't have to wait long for another shot, with Jack Bird's catch in-goal leading to an 80-metre effort which finished with Cody Ramsey putting Mathew Feagai over.
After Jayden Sullivan scored to extend the Dragons' lead, the Titans closed the gap to 22-14 before the break with a converted David Fifita try and a penalty.
Bird scored in the first minute of the second half, and while Jayden Campbell tried to hit back immediately, Dragons skipper Ben Hunt produced a brilliant last-ditch tackle to save a try.
Beau Fermor and Jojo Fifita scored to reduce the Dragons' lead to four with 20 minutes remaining, and when Molo was sent off three minutes later the Titans looked like pulling off a miraculous comeback.
But Moga's two tries in the space of four minutes ended any chance of that happening, with Jack Bird completing his double two minutes from time.
- In addition to scoring three tries, Tautau Moga powered his way to 192 metres and broke eight tackles.
- Despite winning, the Dragons completed fewer sets than the Titans (77 percent compared to 80 percent) and missed the same number of tackles (52) as their opponents.
- Talatau Amone's double means he has five tries in his last two games.
- Phillip Sami was forced from the field in the second minute for a HIA which he failed.
- Francis Molo was sent off for a high tackle on Patrick Herbert with 17 minutes left in the match.
- Dragons centre Moses Mbye left the game on 19 minutes with a hamstring injury and didn't return, while hooker Andrew McCullough was ruled out on 12 minutes with a knee injury.
- The Dragons have won their last four games at WIN Stadium.
- Aaron Woods played his 250th NRL game.
Play of the game
A determined effort for his first try of the day saw Moga stretch out and just touch the try-line. It gave the Dragons the lead after the Titans had assumed a two-point advantage and was an important moment in a free flowing first half.
What they said
"It was very messy, attack was great, but obviously defensively we were too loose for long periods. When we went down a man and had to defend our own end for periods [while] only four in front and did that well. Then we found three tries, so we showed a lot of character there." - Dragons coach Anthony Griffin.
"We did well to keep close to them all game I thought, we were battling away there and then got within striking distance and then just made way too many errors when we got the ball. That made them run away with the game in the end." - Titans coach Justin Holbrook.
The Dragons travel to CommBank Stadium to take on the last-placed Wests Tigers, who they beat 12-6 back in round 8, while the Titans are back at home to play the Knights.
Story first published on NRL.com