The Thrashers squandered a 2-0 lead, but Tkachuk bailed them out when he poked a rebound past Boston goalie Joey MacDonald with 18.6 seconds remaining in the third.
Ilya Kovalchuk and Greg de Vries also scored, and Tkachuk added an assist for the Thrashers, who swept the four-game series with Boston.
Marco Sturm and Petr Kalus scored, and MacDonald made 37 saves for Boston, which has lost seven of eight to drop out of playoff contention.
The Thrashers won their second straight and improved to 41-27-11, tying the franchise record for wins and moving five points ahead of Tampa Bay in the Southeast Division with three games left. The Lightning have four left, including Saturday night against Washington.
Atlanta, which won 41 games last year, can clinch a playoff berth with a win in any of its last three games. The Thrashers would also clinch with a loss by both Toronto and the New York Islanders.
The Thrashers improved to 10-4-1 since acquiring Tkachuk from St. Louis for forward Glen Metropolit and three draft picks on Feb. 25. Before the trade, Atlanta had lost six of seven to tumble from first place in its division to sixth in the Eastern Conference.
Tkachuk has been a key part of a resurgence that has Atlanta on the brink of its first playoff appearance since joining the NHL as an expansion team in 1999-2000. He has seven goals and five assists with the Thrashers.
The Thrashers outshot Boston 16-7 in the first period and took a 1-0 lead 24 seconds in when Garnet Exelby deked MacDonald and fed Kovalchuk for an easy goal.
After de Vries extended Atlanta's lead to 2-0 midway through second, Sturm cut the deficit to 2-1 with 1:30 left in the period when he deflected Zdeno Chara's slap shot into the net.
Kalus tied it 2-2 with 4:57 left in the third, setting up Tkachuk's winner.
Notes: Boston was charged with four first-period penalties, while Atlanta had none. ... Kovalchuk has 199 career goals. ... Tkachuk made his Atlanta debut in Boston on Feb. 26. ... Atlanta outshot Boston 40-27. ... Boston has lost six in a row to Atlanta.