The Seagulls splashed out £300,000 to buy Murray from Rochdale, their most expensive signing for 25 years.
He made his first appearance as a substitute in the midweek defeat at Northampton but Brighton soon saw a return on their money as Murray was given a proper chance to shine in front of the home fans.
He almost opened his account in the 20th minute, gliding past a defender from a Dean Cox pass to shoot inches wide of the far post.
Murray only had to wait three more minutes for his first goal for the club. A Cox corner was met by Guy Butters with a header which a Crewe defender blocked on the line and Murray pounced to find the roof of the net with a deflected drive from close range.
Butters doubled Brighton's lead in the 41st minute. The Crewe defence failed to deal with another corner from Cox and the veteran centre-half, returning from a calf injury, fired in a rare goal.
The game was all over on the stroke of half-time when a cross from Cox was flicked-on with his head by Nicky Forster for Murray to score with an emphatic right-foot volley.
The second half was an anti-climax. Brighton took their foot off the pedal and Crewe, inspired throughout by the transfer-listed Eugen Bopp, could have reduced the arrears.
Bopp released Byron Moore for a shot which Brighton keeper Michels Kuipers saved with his legs and Moore also blazed over when Kuipers pushed a long-range free-kick from Patrick Boyle into his path.
Murray was much quieter after the break, but he almost completed his hat-trick with 13 minutes left. Cox's free-kick reached him beyond the far post and he cut back inside for a shot which was blocked on the line by a crowd of Crewe defenders.