Hartlepool, who have never won at Brighton, were 2-0 down after 17 minutes and have now conceded four goals in each of their last three away games.
Chris Wood, on loan from West Brom, gave leaders Brighton the lead after only 12 minutes by shooting through a crowd of players after Gary Dicker had played the ball into the box from a short corner.
Murray doubled the Seagulls' advantage with a tap-in five minutes later following intelligent play from Ashley Barnes. Murray was unlucky when his shot deflected off the post and the Brighton striker later headed wide following a fine ball to the far post by Elliott Bennett.
Substitute Craig Noone struck Brighton's third eight minutes after the interval from close range, and Murray added his second in the 57th minute with a clinical finish in a one-on-one situation with goalkeeper Scott Flinders.
The Seagulls were caught ball-watching when Pools midfielder Antony Sweeney grabbed a consolation for the visitors in the 63rd minute on the edge of the six-yard box.