Playing without their injured Netherlands international center Peter van Paassen, Amsterdam struggled inside.
Proteas, who hosted last year's EuroCup Final Four, are now just a victory away from making it to the showpiece event again.
They will go for a 2-0 sweep on Thursday when Game 2 is played in Holland.
Bosnia and Herzegovina international Haris Mujezinovic scored 17 points and pulled down a game-high nine rebounds for Dragan Raca's Proteas outfit - one of the class acts in this competition.
Also for the winners, Vujadin Subotic weighed in with 16 points and Quincy Taylor 14, including 12 in the second half.
Teddy Gipson's 17 points and Clint Harrison's 16 led Amsterdam.
Sergio de Randamie's lay-up for the visitors accounted for the last points of the second quarter and sent the teams to the changing rooms with Proteas clinging to a 39-37 advantage.
Robert Krabbendam then buried a jump shot after the restart to knot the score at 39-39.
Proteas went back in front by five after a Goran Nikolic three-point play, only for Amsterdam to come back again with a Harrison three-ball tying the game at 46-46 midway through the quarter.
The hosts finished the frame on a 16-7 run, capped by Darrel Mitchell's alley-oop pass to Taylor and they went to the last frame on top 62-53.
Taylor opened the fourth quarter with a lay-up and Subotic drilled a three as Proteas opened up a 67-43 lead - their biggest of the game.
Amsterdam, despite the absence of their big center Van Paassen, fought brilliantly and twice clawed the deficit back to five at 75-70 with 2:14 remaining and 77-72 with just 1:40 left.
Raca decided to call a timeout to discuss the end-of-game strategy.
When play resumed, Subotic missed an attempt from three-point land and Amsterdam had a chance to pull even closer but Orien Greene lost control of his dribble and committed his fifth turnover of the game.
Mujezinovic gave it right back with a bad pass, however, the Dutch team wasn't able to take advantage.
Clint Harrison was off target with two three-pointers and Proteas ended up padding their lead at the free throw line.
Proteas won the battle of the boards 32-25 and they also excelled inside the arc, making 24 of 40 (60%) compared to 17 of 32 (53.1%) for Amsterdam.
Harrison finished four of eight from long range and Gipson was three of five.