Webster, who was starting for Portsmouth for the first time this season after a successful loan spell at Aldershot, tapped into an unguarded net in the second minute.
The right-back latched onto a loose ball in the six yard box, after Andy Rafferty had saved Ryan Taylor's header from Ricky Holmes' deep free-kick.
The home side had a fantastic chance to double their lead a minute later, when Wes Fogden was tripped in the box by Christian Burgess.
But Taylor's penalty was well saved by Rafferty, going down to his right, before Holmes failed to keep his rebound down.
Danny Hollands had a chance to give Portsmouth a 2-0 half-time lead but his strike was deflected for a corner, after an impressive solo run.
The second period lacked quality goalscoring chances, although Luke James forced Trevor Carson into his first save of the game, but the keeper got down well to deny Hartlepool's leading goalscorer.