A second-generation professional baseball player, Joe Biagini has long wanted to make his father proud by reaching the big leagues.

Sunday, Biagini realized that dream with his first day on a major league roster, as the right-handed pitcher was included on the Toronto Blue Jays’ opening-day roster.

While Biagini didn’t pitch in the Blue Jays’ 5-3 win in Tampa Bay, he was still floating on air in waiting for the feeling of reaching the big leagues, after spending four seasons in the San Francisco Giants’ farm system, to set in. He said it still hadn’t after receiving the news last Wednesday he was Toronto bound.

“I’ve been waiting for like 20 years to have that phone call with my dad,” Biagini said. “So it was pretty special to say the least.”

As a Rule-5 draft pick — a type of offseason waiver-wire selection process that allows organizations to pluck players from other teams’ systems if not included on the 40-man roster after a certain number of years as a pro — Biagini will have to play the entire season in the Major Leagues or be offered back to the Giants.

Getting the opportunity to make the big league jump was bittersweet for Biagini. A Sunnyvale native who graduated from The King’s Academy — he also played for one year at the College of San Mateo in 2009 before transferring to UC Davis — Biagini grew up a Giants fan. His father Bob also played in the Giants’ farm system for one of his two pro seasons.

The Giants selected Biagini out of UC Davis in the 26th round of the 2011 draft. He went on to post a 29-30 career record, topping out at Double-A Richmond last season where he notched a 10-7 record with the second-best ERA in the Eastern League at 2.42.

“I really enjoyed the Giants organization,” Biagini said. “It was a good group of people. … They definitely preach the family environment, which I like playing in. The Giants definitely gave me a lot of opportunities because I struggled through my first couple years … but they kept believing in me and kept letting me start.”

After serving as a starting pitcher his entire career — he has tabbed three pro relief appearances, two coming in his first season of 2012 at Low-A Salem-Keizer — Biagini will start the year in the Blue Jays’ bullpen. Among Toronto’s relief corps, he joins former Stanford closer Drew Storen and former Oakland A’s pitcher Jesse Chavez.

“The coaches to me have been really encouraging in communicating all the basics and allowing me to move into this role,” Biagini said. “I definitely feel good about my progress and feel I can contribute to any role I am put in.”

Biagini posted a solid spring in the Grapefruit League. Through seven appearances, he logged nine innings, allowing four runs on eight hits while striking out eight against three walks. Known as a sinker-slider guy, he added velocity on his fastball. He touched 95 mph in at least one outing, according to the Toronto Sun.

Still, Biagini said he didn’t concern himself with overthrowing to get results. Instead, he stayed within himself and hoped his potential would shine through any bumps along the road.

“Obviously the team wanted to see success from me,” Biagini said. “But it seemed like they had an understanding of what my stuff can be and how it will translate. If I show my ability to keep the ball down and get some groundballs … they were just looking to see if I was the guy they thought I was coming in. I had some good velocity in the spring as well so I think that helped solidify what I was working on.”

Being a Rule 5 draftee can be a difficult path though. Last season, former Santa Clara University standout J.R. Graham was a Rule 5 selection of the Minnesota Twins. The Livermore native endured an entire season on the Twins’ roster, posting a 4.95 ERA through 39 appearances. This season, however, Graham was optioned to Triple-A Rochester to start the year.

Through the spring, and even for Sunday’s opener on the road, Biagini sported the No. 66. He said he intends to ask for a different number when the Blue Jays return to Toronto for Friday’s home opener against the Boston Red Sox.

Not that he has a problem with the unorthodox baseball number, per se. He said not wanting to wear No. 66 is more out of respect for former Blue Jays infielder Munenori Kawasaki — a popular player through his three years in Toronto — who wore that number before departing via free agency after last season.

“Out of respect for the previous owner, I’m going to try to change it to something else,” Biagini said.

It’s just one of many things on Biagini’s to-do list, which includes finding a place to live in Toronto, not to mention adjusting to living in Canada, the first time he’s lived in another country. But he’s first and foremost focused on the job at hand — helping the Blue Jays get back to the postseason, where they were defeated in the American League Championship Series by the eventual world champion Kansas City Royals.

“It’s a very welcoming group and it seems like everyone contributes something different,” Biagini said. “I’m still getting to know them but I’m just trying to take in as much as I can.”