Goverla's fans were already celebrating the fact that captain Dante Swanson had knifed his way into the lane, drawn a foul and put the team on top, 75-74, with two free-throws.
But Prostejov inbounded the ball to Smith and as he crossed midcourt near the scorer's table, he dribbled behind his back and drifted to the edge of the three-point line before leaning in and releasing the game-winning shot.
To his credit, Swanson was guarding Smith as close as possible without fouling him.
It was a remarkable finish that saw Smith turn around, run towards his team's bench and chest-bump teammates.
"It is very difficult to lose like this, in the last seconds," Goverla's Alexander Shostuha said.
"A crazy shot of Smith decided an interesting game."
Crazy or not, Smith showed he's lethal from long range because he buried four of his seven three-ball attempts.
The Czech Republic's other team in the EuroChallenge, BK Pardubice, are in Group C and they also got off to a winning start with an 82-77 home triumph over Roanne.
Pardubice also won a thriller thanks to Maurice Hampton, who had 25 points.
His biggest shot was a go-ahead three-pointer with 47 seconds to play.
Hampton drilled five of seven shots from long range.
Both of Hungary's teams won their opening games, including Atomerömü SE.
The Belgian team tried to claw their way back from a 19-point deficit and closed the gap to 69-68 when Nate Fox connected from behind the arc in the fourth quarter.
Kulagin coolly went to the other end and nailed a three-ball of his own to calm his team's nerves.
He made a real impression on Brad Dean, the Okapi coach.
"He made his shots in the key moments, when his team needed to make the run," Dean said.
Triumph had to go through the Qualifying Round.
SENE BACK ON THE BOARDS
Mouhamed Sene of Senegal was a very useful player for Gravelines Dunkerque last season in the EuroChallenge when he averaged 10.8 points and 8.2 rebounds.
A 2.11m center, Sene was so good that he had three straight double-doubles in one stretch against Lisboa e Benfica (15pts, 13rbs), BK Ventspils (15pts, 10rbs) and again against the Portuguese, Benfica (16pts, 12rbs).
Now on the books of Fuenlabrada, who already have the physical frontline presence of Mexican center Gustavo Ayon, Sene began the 2011-12 EuroChallenge with, you guessed it, a double-double.