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.
"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."
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