Suvorov, who came on for Nicolae Josan in the 58th minute, scored from a free-kick in the 69th minute of the first match between the two nations in nearly 12 years.
Doros's long-range goal in the 74th minute secured Moldova their first win in a competitive match in nearly three years.
Finland captain Sami Hyypia was sent off in the 36th minute for fouling Viorel Frunze when he had a clear goal scoring opportunity.
"We played to win from the beginning and our tactics worked," Moldova coach Gavril Balint said. "We deserved the victory."
Finland coach Stuart Baxter said the game "changed completely" after Hyypia's dismissal "and the team was unable to regroup."
Midfielder Roman Eremenko added: "We wanted to win but the loss should not be treated as a tragedy. There are plenty of matches to come."
One of the poorest countries in Europe, Moldova had not won a competitive match since a 3-0 victory over Hungary in November 2007.
Moldova will be eager to keep up their winning run against Hungary in their next qualifier on Tuesday, while Finland will be hoping to put up a better performance when they travel to the Netherlands to face the World Cup runners-up.
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