Triumph guard Marque Perry's lay-up capped a 10-2 run to knot the score at 90-90 at the end of regulation, but Darius Vashington's three-point play in the final seconds of the extra period secured a 102-100 road win for Ural Great.
The second game in the best-of-three series will be played in Perm on Thursday.
Ural Great appear to be undergoing an amazing transformation.
The club had been struggling but new Serbian coach Aleksandar Kesar has helped instill self-belief in the squad since taking the helm two weeks ago.
At the weekend, Ural Great upset high-fliers BC Khimki in Russia's Superleague and now they have moved within one victory of the EuroChallenge Final Four.
Vashington led the Ural Great scorers with 29 points, including nine in the extra period.
Even in overtime, though, Ural Great had to sweat out a nervous finish.
Vashington's two free throws had pushed the visitors into a 98-95 advantage with 22 seconds to play in OT, but Ralph Biggs stopped the clock with an ill-timed foul on Ernest Bremer just three seconds later.
The American went to the charity stripe and made both attempts to close the gap to 98-97 and after Vashington sank just one of two free throws, Perry drilled a three-ball with 12 seconds to go.
That gave Perry 26 points, and made it 100-99 in favour of Triumph.
Vashington replied with a lay-up six seconds later, however, and was fouled on the play by Bremer.
He made the free throw for the final point of the game.
Triumph did go in front by as many as 12 points in the second quarter, but Stanislav Eremine's team did not press home the advantage.
Instead, Ural Great went on a 14-4 run to cut the deficit to 32-30 and they eventually took a 41-39 lead to half-time after a basket from Fedor Likholitov.
Vashington had 14 points at the break while Perry led Triumph with 10.
Ural Great took control in the fourth quarter, only to suffer a late collapse.
In overtime, they did just enough to claim a valuable road win.
One of the key men for Ural Great was Vanja Plisnic, who made four of five from long range on his way to 20 points.
He also had 10 rebounds.
Bremer had 17 of his 20 points after half-time for Triumph.
Ural-Great Perm coach Aleksandar Kesar:
"I want to congratulate my players, because we played without some very important players - Bashminov (injury) and Surovtsev (stomach problems). It held us at a disadvantage, but we still led by as many as 11 points in the game. We were not smart at the end of regulation, but we deserved this victory. I'm very happy that we won in Lyubertsy."
BC Triumph player Valery Likhodey:
"In my opinion we lacked concentration in this game. We let our rivals to shoot, and they succeeded in that department. It was very hard to stop them, and only at the end of the game we tied the score and took the game to OT. But we made some mistakes in defense and lost by two points. We hope to show our best game on the road. We have to win in Perm."
BC Triumph coach Stanislav Eremine:
"We failed to stop Vashington - he is the key player of Ural-Great."