Team Sky's Viviani triumphed on the 177-kilometre ride from Albenga to Genoa, pipping Moreno Hofland and Andre Greipel on the line.
"It's incredible to win my first stage in the Giro. We're here to help Richie Porte in the general classification, so I don't have many riders for the sprints," he told reporters.
"But I knew the final metres were uphill and I knew when to go."
Matthews, whose Orica-GreenEDGE outfit won the opening team time trial on Saturday, finished seventh on the stage to take the pink jersey.
"It was a very technical stage finish and we knew that you had to be up front in the last kilometre. I decided to go early, because everyone was fighting for position. But the finish was really hard," he said.
"It was a really special time for me to wear the Maglia Rosa last year, and to wear it for the second time is even sweeter. It was team work yesterday that got me where I am today, so I have to thank my team mates for this magical feeling."
Spanish favourite Alberto Contador is sixth overall for Tinkoff-Saxo as he bids to become the first man to achieve a Giro/Tour de France double since the late Marco Pantani of Italy in 1998.
Stage three on Monday is a 136-km trek from Rapallo to Sestri Levante with several tough climbs and bends. The world's second greatest stage race finishes in Milan on May 31.
(Writing by Mark Meadows, editing by Ed Osmond)
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