Doubles specialist Siniakova and Machac, who are in a relationship off the court, combined superbly and held their nerve in a tense finale to win 6-2, 5-7, 10-8.
Canadians Felix Auger-Aliassime and Gabriela Dabrowski beat Dutch team Demi Schuurs and Wesley Koolhof to take the bronze.
The Czechs hardly put a foot wrong in the first set but Zhang and Wang, the second Chinese pair to reach an Olympic doubles final, raised their level in the second, pushing the match into a superb tiebreak.
"Super tight breaks are tough because they can go either way but I think the mindset was really good, we fought until the end," Siniakova, a nine-times Grand Slam doubles champion, told reporters.
The Czech also won a women's doubles gold medal on her Olympic debut in Tokyo.
"I don't like to compare because every trophy, every medal is really amazing and has different emotion, different feelings," she said. "I am proud and I'm happy that we could achieve the medal and we got the gold."
Machac will seek another medal on Saturday when he pairs up with Adam Pavlasek against Americans Tommy Paul and Taylor Fritz for the men's doubles bronze.