Matthews and Mads Pedersen were neck-and-neck until the final corner, but the Australian, who was set up perfectly by his Jayco-AlUla team, moved in front on the home straight to emerge victorious and claim a first stage win at the Giro since 2015.
"We rode all day today. They were fully committed for me to win the stage," Matthews said.
"I don't have words at the moment. It's been such a roller coaster this year. Now already on stage three with a stage win? That's more than I could ever dream of."
Trek-Segafredo's Pedersen finished second, ahead of Kaden Groves (Alpecin–Deceuninck) and Vincenzo Albanese (EOLO-Kometa).
Corratec-Selle Italia riders Alexander Konychev and Veljko Stojnic were the early pace-setters, leading the stage for the first half of the race, before being reeled in by the main group with 36 km to go.
Evenepoel in the lead
Belgian Remco Evenepoel extended his lead after taking three bonus seconds to tighten his grip on the leader's maglia rosa jersey, while Joao Almeida recovered from a fall in slippery conditions to move up to second.
Primoz Roglic, who was eclipsed by rival Evenepoel in the intermediate sprint, climbed to third.
"I went for the time bonus in the intermediate sprint because we were just there as we wanted to stay at front because the rain had made the road a bit dangerous," Evenepoel said.
"I saw Roglic and his teammates right behind us so I sprinted. It didn't cost much energy. There were a few seconds up for grabs. One second more on Primoz and three on the others, it's good, especially after such an easy day of racing."
Tuesday's stage four is a 175km ride from Venosa to Lago Laceno, which features two long and manageable ascents before culminating in a steep climb and a flat finish.
See the overall standings and follow the race on Flashscore.
