Pellegrini, brought on just after the hour-mark with the score 1-1 in Yerevan, fired his side ahead in the 77th minute - to break the resistance of the home side.
Andrea Belotti then scored his second of the game moments later to wrap up a valuable three points in the race to qualify for the 2020 European Championships.
"My first thought after scoring was for the girls watching me at home; my wife and my daughter," said Pellegrini afterwards.
"It’s a great night for me personally, and I'm happy we won a game that was not simple - Armenia are tough, especially on their own turf."
Roma captain Alessandro Florenzi started the match, while Henrikh Mkhitaryan captained Armenia. Both played the full 90 minutes.
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