Smoak, who entered the game batting .077 in 11 exhibition games, connected off starter Burch Smith in the third for a two-run drive. Smoak followed in the fifth with his shot that knocked the Rays' offices, off Ernesto Frieri.
Norris, one of Toronto's top prospects, yielded five hits, walked one and allowed a run on a sacrifice fly by Evan Longoria in the first.
"I thought he was fantastic today, but he's good," manager John Gibbons said. "The key is his confidence and starting to stretch some things out a little bit. He's getting some innings under his belt and the sky's the limit. We'll see where it takes him."
Ryan Goins added a two-run homer in the fifth for the Blue Jays.
Rays: Smith, a contender for a spot in the rotation, allowed six runs on six hits in 3 2-3 innings. The right-hander also walked three and struck out three on 63 pitches.
"I think I did some stuff well," Smith said. "I felt like I threw some good off-speed pitches. I'm glad I got my pitch count up."
Blue Jays: RHP Marcus Stroman had season-ending surgery Thursday to repair the torn anterior cruciate ligament in his left knee. Stroman was not available for comment, but tweeted: "Surgery successful. Thank you all for the well wishes. I feel amazing!" ... INF Maicer Izturis left in the second inning with a groin strain.
Rays: OF Brandon Guyer (strained oblique) started in centre field and went 2 for 3. ... LHP Drew Smyly (shoulder tendinitis) played catch from 100 feet on Thursday and is expected to throw off a mound on Saturday. Manager Kevin Cash said he wants Smyly to throw three or four times before being reevaluated. ... LHP Jake McGee (elbow surgery) threw off a mound for the second time in three days without discomfort. ... C Rene Rivera (mild calf tightness) will take a couple days off to rest as a precaution.
STILL NO STARTER
With ace Alex Cobb sidelined for the near future because of right forearm tendinitis, Tampa Bay is still in search of a starter for its season opener against Baltimore on April 6. But Cash isn't rushing to name one.
"We're not naming anybody 'til Buck does," the first-year manager quipped, referring to Orioles manager Buck Showalter.
All jokes aside, Cash said he wants an opportunity to examine all of his options before officially jumping to a decision. Right-hander Chris Archer, 10-9 with a 3.33 ERA in 32 starts last season, seems to be the likely choice.
"We're trying to be cautious and not disrupt people's plans and kind of what they're on-track for," Cash said. "So, there's a lot of moving parts right now. ... We're trying to factor in all that and we just haven't gotten to exactly what the best scenario we want to go with."
After attempting to make a diving catch in the fourth, Rays OF Steven Souza rolled on the field while clutching his right arm in pain. After being checked out by trainers, Souza opted to stay in the game, but was diagnosed with forearm tightness and is scheduled to have a precautionary MRI.
Blue Jays: R.A. Dickey will make his third exhibition start on Saturday against the Phillies in Clearwater. The knuckleballer gave up two runs in a split-squad game against the Canadian junior national team on Sunday.
Rays: Jake Odorizzi will start on Saturday against the Twins in Port Charlotte. The righty has allowed just one earned run in three outings this spring.