The deal is worth $38.5 million, a source told ESPN.com's Pierre LeBrun Thursday.
Myers, 21, will receive a $10 million signing bonus July 1, 2012, the first day of the extension, the source said. He will receive $2 million in salary next season and $6 million in 2013-14.
In 2014-15 and 2015-16, he will receive $5 million each season, followed by salaries of $4 million, $3.5 million and $3 million, the source said.
Sabres general manager Darcy Regier said the contract talks lasted just two weeks.
"You want to avoid contentious negotiations, and create a relationship that is a win-win situation," Regier said during a news conference Thursday in Buffalo. "We wanted to recognize Tyler for his abilities now, and in the future."
Myers, the NHL's Calder Trophy winner as rookie of the year in 2009-10, would have been a restricted free agent July 1.
Myers, 6-feet-8, 222 pounds, was the 12th overall pick in the 2008 draft. He finished with 10 goals and 27 assists last season for 37 points, sixth-best on the team.
Buffalo already made several high-profile splashes this offseason.
The team traded for the rights to defenseman Christian Ehrhoff, then signed him to a 10-year, $40 million contract a day before the start of free agency.
The Sabres opened free agency by signing dependable two-way forward Ville Leino to a six-year, $27 million contract.
Those moves contributed to putting the Sabres payroll about $5 million over the NHL salary cap, which is allowed during the offseason. Regier has until the start of the season next month to get the Sabres' payroll below the cap.
Information from ESPN.com's Pierre LeBrun and The Associated Press was used in this report.