Ansu Fati scored a brace and Gavi added the third in what was also the last game at Spotify Camp Nou for the foreseeable future, with Barca due to move to the Olympic Stadium next season to allow their iconic home to be redeveloped.
However, the night belonged to club captains Busquets and Alba as they played their final ever games for Barca, with coach Xavi Hernandez later confirming they will not be involved in the final fixture of the season at Celta Vigo next week.
Alba, 34, broke down in tears when he was taken off late in the second half to a standing ovation, hugging all his teammates individually as the supporters chanted his name.
Busquets, also 34, wore a smile as he followed the left-back off the pitch a few minutes later, handing the armband to Sergi Roberto.
"I dreamed of one day playing in this stadium," Busquets said after the game in a speech addressed to the 88,775 fans present at Camp Nou.
"When I was a kid, I came here to watch matches, I went to away games, I watched on television. Nobody will ever take away the feeling of pride of belonging to the best club in the world.
"I am leaving, but the dream I had has been fulfilled -- in fact, it's been better than I ever imagined. This is also not a goodbye, it is a 'see you soon.'"
Alba added: "Thanks for the affection you have shown me in the street, at the stadium ... everywhere. I have been proud to wear this shirt. This has been one of the happiest days of my life.
"I've played less than usual this season, but I've tried to help everyone. Now it's time to see things from a different perspective, but you have a friend here for life."
Busquets announced earlier this month he would not renew his contract with the club, despite coach Xavi Hernandez making it clear he wanted the midfielder to stay for another year.
The former Spain international brings to an end 18 years with the Catalan side, during which time he has made over 700 first team appearances, winning nine LaLiga titles and three Champions Leagues.
Alba, meanwhile, leaves after 11 years in the first team. After being released from the academy, he returned in 2012 and has gone on to play over 450 times for the club, winning six LaLigas and one Champions League.
Neither have yet announced where they will play their football next season.
Barca also said a temporary farewell to Camp Nou, with the 99,000-seater ground to be revamped over the next 18 months, forcing the team to play their home games at the 50,000-capacity Olympic Stadium in the Montjuic area of the city until at least November 2024.
After the game, following the tributes to Busquets and Alba, Barca played footage of historic moments that have taken place at the stadium since its opening in 1957.
Several featured Lionel Messi, prompting chants of "Messi" from the fans as the Argentine forward continues to be linked with a move back to the club this summer with his Paris Saint-Germain contract expiring.
A fireworks display then took place and confetti was let off as music blared out and the curtains came down on Camp Nou in its current guise.
"In a year and a half, we will have the best stadium in the world," Busquets added. "We will have a home in line with the magnitude of this club and its history."