Anssi Salmela and Zach Bogosian also scored for Atlanta.
The Thrashers scored the first four goals before Tampa Bay, led by two second-period goals by Martin St. Louis, tried to rally. Andrej Meszaros scored on a tip-in to cut Atlanta's lead to 4-3 early in the third period.
Atlanta ended the comeback as Peverley and Kovalchuk each scored their second goals.
The 22-year-old Pavelec earned the starting job in the preseason as Kari Lehtonen continues to recover from offseason back surgery.
Mike Smith had 22 stops for Tampa Bay.
Atlanta's 18-year-old Evander Kane, the fourth overall pick in the June draft, set up Peverley with a pass from the slot to the right circle, giving the Thrashers a 4-0 lead.
Tampa Bay's Victor Hedman, the second overall pick, earned his first assist on the first goal by St. Louis.
St. Louis scored again on a power play with 45 seconds remaining in the second period to cut Atlanta's lead to 4-2.
Tampa Bay drafted the 6-foot-6 Hedman and acquired Mattias Ohlund, Matt Walker and Kurtis Foster in the offseason in hopes of improving its defense.
The Thrashers took a 2-0 lead in the first period on goals by Kovalchuk and Salmela. Smith deflected Kovalchuk's shot, but Tampa Bay's Paul Ranger's stick hit the puck into the net.
Bogosian, the third overall pick in 2008, scored from the left circle midway through the second period. Peverley earned an assist on Bogosian's goal before scoring for the 4-0 lead.
The game was the first season-opener for Rick Tocchet as Tampa Bay's coach. He was named coach on Nov. 18, 2008.