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.

Some 3,000 people have been left homeless, according to a BBC reporter who has been to the scene.
Their houses were destroyed after mud and rocks, dislodged by the rain, plunged down the hillside.
Reports from the Democratic Republic of Congo say there have been similar problems across the border.
UN-backed Radio Okapi says 14 people have been killed on the Congolese side of Lake Tanganyika.
Burundi's President Pierre Nkurunziza has visited the scene of the landslides, 35km (20 miles) south of the capital Bujumbura.
The governor of Bujumbura district, Jacques Minani, described it as "a disaster", reports the AFP news agency.
He said that the emergency services were now searching the area for survivors.

Robinson, 33, who was released by relegated Bolton Wanderers at the end of last season, has signed a one-month deal that runs until 26 October.
Clark said: "I'd like to thank the board for allowing me to bring in Paul.
"He is an experienced defender who will enhance our playing squad."
Already without injured skipper Stephen Carr, who has a career-threatening knee injury, Clark last week lost season-long loan signing Paul Caddis, who is expected to be out until Christmas at the earliest once he has had shoulder surgery.
The injuries have left us with just two fit senior defenders
And, after losing David Murphy with a torn hamstring midway through the first half of Saturday's 5-0 home defeat by Barnsley, the Blues boss is now also without Pablo Ibanez, who is expected to be out for four weeks after breaking a bone in his foot.
"While I have been impressed with the progress of our young defenders, the injuries have left us with just two fit senior defenders," added Clark. "So the acquisition of Paul was one we needed to make.
"As a senior professional, Paul will give us that added experience which is always useful in this competitive division."
Former England Under-21 international Robinson began his career with home-town club Watford before being signed by Gary Megson for Albion in 2003.
After six years at The Hawthorns, a season-long loan move to Bolton was made permanent in 2010.
But Robinson, who is expected to make his Blues debut in Saturday evening's clash with leaders Brighton at the Amex Stadium, has not played since finishing last season out on loan with Leeds United.

The South Africa-born 21-year-old right-hander will now remain at Lord's until the end of the 2017 season.
Eskinazi was a regular for Middlesex's Second XI this year and made his first-class debut in last month's victory over Yorkshire at Lord's.
"Stephen thoroughly deserves a new contract," managing director of cricket Angus Fraser said.
"Once again he was our top run-scorer in the second eleven and was rewarded for this with a first-class debut.
"He looked comfortable against high quality bowling, which is a good sign for the future."
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.
[NEXT_CONCEPT]
At least 10 people are missing, feared dead, in Burundi after landslides which have destroyed hundreds of homes, officials say.
[NEXT_CONCEPT]
Birmingham City boss Lee Clark has responded to his side's string of defensive injuries by bringing in former West Bromwich Albion left-back Paul Robinson on a short-term deal.
[NEXT_CONCEPT]
Middlesex wicketkeeper-batsman Stephen Eskinazi has signed a new two-year contract with the club.