Pedro Alvarez doubled twice and scored both runs for Pittsburgh.
Rasmus hit an RBI single in the first inning off Gerrit Cole. Navarro's double scored Rasmus.
Cole struck out two and gave up three hits in two innings.
After spending last season with the Chicago Cubs, Navarro signed a two-year, $8 million contract with the Blue Jays in the off-season.
Navarro, who batted .300 with 13 home runs for the Cubs, said he has been taking extra swings to improve at the plate. His main focus, however, is getting to know the pitching staff.
"Every time I get the opportunity, I jump in the cage. But right now it's all about the pitchers," Navarro said. "Especially with me coming to a new team (and) seeing a lot of new faces. It's all about them. I will get my swings in. I will do my work."
Blue Jays starter Mark Buehrle allowed one earned run in two innings. He gave up two hits, one walk and struck out one.
Blue Jays outfield prospect Ricardo Nanita hit an RBI triple in the eighth.
Matt Hague hit an RBI single and Gaby Sanchez doubled for the Pirates.
Toronto star Jose Bautista, who homered and doubled twice in the first two exhibition games, was given the day off.
Blue Jays: Buehrle threw 43 pitches. He said his command was good overall, but he felt like he had trouble keeping his sinker in the strike zone.
"I wasn't happy with how sharp it was today," he said. "I wasn't throwing too many sinkers for strikes. But it's the first outing. Things are going to turn and get better as we go along."
Manager John Gibbons disagreed and said the left-hander looked better than he did during camp last season.
"I thought Buehrle for his first start was really good," Gibbons said. "I told him I thought he was a little sharper in this camp than he was (at the same point) last year. It showed today, that's a good sign."
Pirates: Cole had a rocky first inning, giving up three hits and two runs. But the right-hander settled down in the second, striking out Dan Johnson and Ryan Goins and getting Maicer Izturis on a grounder.
"I don't overcook their first outing," Pirates manager Clint Hurdle said of his pitchers. "They need to get out there. They need to reacquaint themselves with adrenaline, mound, rhythm and all that."
"I thought Cole's second inning was much more effective, just in terms of downhill plane, getting the ball downhill, mixing in the breaking ball."
HAPPY TO BE BACK
For the first time since sustaining a season-ending forearm injury last May, right-hander Brandon Morrow appeared in a game with the Blue Jays. Morrow allowed two hits and one run in two innings. He said he felt comfortable on the mound and was happy to finally see game action.
"I felt great. I felt on time with my delivery," he said. "I was most happy with the life I had on my fastball and did what I wanted to."
Navarro said he felt Morrow had strong command of his pitches, despite giving up a run.
"He kept the ball down good, threw a couple good sliders and a couple good splitters," Navarro said. "I know people want to see zeroes on the board, but these guys are locating their fastball and then working off the fastball. As spring training goes on, velocity is going to go up."
Sergio Santos was scheduled to appear in relief for the Blue Jays on Friday, but the right-hander was told to stay home after getting a stomach virus on Thursday night.
Santos, who was 1-1 with a 1.75 ERA in 29 appearances last season, is expected to return to the Blue Jays on Saturday. His first outing has been rescheduled for March 4 against Philadelphia in Clearwater.
Chad Jenkins took Santos' place and allowed no hits in one inning of relief.