The visitors started the game with a 5-0 run, but the hosts quickly closed the gap and scored 9 unanswered points in the second quarter to build an 11-point lead.
Trailing 49-38 late in the second, Tartu made three consecutive 3-pointers to cut it to 49-47.
Peciukevicius's basket tied it at 49, but Jeremy Chappell's buzzer-beating bank shot put Avtodor up by two at halftime. The teams combined for 63 points in the second period.
Things remained close in the third as Tartu led 69-67 going into the final quarter. Joosep Toome hit a three that gave the visitors a 94-91 lead with 34 seconds to go in regulation, but Micah Downs answered with his own triple to send the game into overtime.
In the extra period, Janar Talts hit a decisive 3-pointer with one minute remaining to give his team the lead back, and Avtodor turned the ball over on its next possession.
Tartu missed a three at the other end, but Peterson's final 3-point attempt fell short.
Downs scored a season-high 22 points and had team-highs in rebounds (7) and steals (4).
Travis Peterson finished with the same number of points, adding 6 boards.
Three other players also scored in double figures. Courtney Fortson had 15 points, 9 assists and 6 rebounds while Alexey Vzdykhalkin and Artem Klimenko scored 10 apiece. The American guard played all but one second of the game's 45 minutes.
Tartu converted 19 3-pointers and shot an impressive 57.6 percent from long range.
Gert Dorbek, Janar Talts, Augustas Peciukevicius and Joosep Toome went a combined 17-for-25 from beyond the arc.
Avtodor is now 8-3 at home across all competitions.