Matthews spent most of the day in a five-man breakaway before sprinting to the line at the end of the up-and-down 138 km route, which started in Ogden and featured 2,174m of climbing.
Garmin-Sharp's Christian Vande Velde retained the overall lead of the race.
"I tried to follow (Team Exergy rider) Freddie Rodriguez because I heard he's a pretty good sprinter," said Matthews, whose team left their own lead-out sprinters due to the mountainous nature of the race.
"So I followed him for a bit and then it got a bit messy. A couple of guys came underneath on the last corner with about 800 to go. I lost a couple positions there, but (UnitedHealthcare's) Rory Sutherland started moving up with 300 to go, so I just followed him and started my sprint at 250 metres.
"I had to go around the long way, but luckily I still had the legs for the finish."
Friday's stage four sees the peloton return to a flatter ride between Lehi to Salt Lake City, ideally suited for sprinters.