Anthony Knockeart converted Gaetan Bong's fine cross to give Brighton the lead in an otherwise even first half.
Bournemouth loanee Murray doubled the lead from close range after the break and he tapped in his second late on.
Defeat was Forest's first of the season under new head coach Philippe Montanier, who joined in June.
Relive Brighton's win over Nottingham Forest
Adding victory to their draw at Derby on the opening day, the result saw Brighton go to the top the table ahead of the rest of the weekend's fixtures.
Murray returned to the Seagulls on a season-long loan this summer and his second-half goals made the points safe as Brighton, who finished third in the Championship last season, took control after half-time.
Tomer Hemed went close to adding further goals for the hosts, while Czech Republic winger Jiri Skalak had a goal correctly ruled out for offside.
Forest have now conceded three goals in both of the first two league games, while Brighton are yet to concede this season.
Glenn Murray scored the first two goals of his second spell at Brighton & Hove Albion as they beat Nottingham Forest to claim a first league win of 2016-17.