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|Andrea Belotti bags a brace. (AFP/Karen MINASYAN)|
Roberto Mancini's side had trailed after 11 minutes in Yerevan when Aleksandre Karapetyan got just the second goal past Gianluigi Donnarumma in the Italy goal in five qualifying games.
But the forward, who plays for Russian club Sochi, went from hero to villan when he was sent off for a second yellow card just before the break to leave his side a man down.
Torino captain Belotti got Italy back level on 28 minutes and added a second 10 minutes from time after substitute Lorenzo Pellegrini had put the away side in the lead his first international goal.
"The satisfaction is to have won a game that was not at all easy," said Mancini, who becomes the first Italy coach to win five consecutive European qualifiers in one group.
"Playing the 11 against 10 paradoxically penalised us because Armenia focused even more on defending and reduced the spaces further.
"But we had the patience to continue to attack even at the risk of exposing ourselves to counter-attack."
Four-time world champions Italy have put their failure to qualify for last year's World Cup behind them and are top of Group J with maximum 15 points, and on course for the Euro 2020 opening match at home in Rome next June.
Armenia are third in the group with six points from five games, behind Finland who have nine points from four games.
Mancini dropped young guns Moise Kean and Nicolo Zaniolo as punishment for being regularly late during the Under-21 European Championships at home in June.
Belotti was joined up front in an attacking trident with Juventus forward Federico Bernardeschi and Fiorentina's Federico Chiesa.
With Juve captain Giorgio Chiellini out with cruciate ligament knee injury, the Armenians piled the pressure on for the first 10 minutes with a slip-up by Nicolo Barella allowing Karapetyan to break through for the opener.
Belotti got the Azzurri back on track following good work by Chelsea defender Emerson Palmieri following a corner with Belotti meeting the curling cross to prod into the net.
Bernardeschi hit the post just after with Belotti also missing a chance before the break as he failed to connect with a Chiesa cross.
But Italy got off the hook when Karapetyan received a second yellow card for a high elbow on defender Leonardo Bonucci.
Mancini made changes around the hour mark with Chiesa making way for Lorenzo Pellegrini and Stefano Sensi replacing Barella.
Pellegrini nodded in after with 13 minutes left, and three minutes later Belotti doubled his tally following good work from Sensi and a bit of luck with a deflected shot which bounced off the Armenian goalkeeper and into the net.
Italy will be looking to extend their run at second-placed Finland on Sunday with Armenia hosting Bosnia-Herzegovina.
Euro 2020 qualifying results:
Republic of Ireland 1 Switzerland 1
Gibraltar 0 Denmark 6
Romania 1 Spain 2
Norway 2 Malta 0
Faroe Islands 0 Sweden 4
Israel 1 North Macedonia 1
Armenia 1 Italy 3
Finland 1 Greece 0
Bosnia & Herzegovina 5 Liechtenstein 0