The victory means the Pilgrims move up to seventh in the Championship, whilst the Saints stay in eighteenth, just six points above the relegation zone.
Plymouth took the lead after 31 minutes when Halmosi chipped the ball past Kelvin Davies in the Southampton goal.
And two minutes later, the lead was doubled. Paterson curled in a long-range effort for his first Plymouth goal since joining from Motherwell in January.
In the second half, Stern John and Wayne Thomas both wasted good opportunities for the Saints and the Pilgrims held on for the win which takes them just two points off the play-off places.
After the game, new boss Pearson admitted that there was definitely room for improvement.
"There's plenty of room for improvement and I would have much preferred to get off to a winning start but it's important we don't sulk about that and get too despondent."