Alba's unstoppable volley at the back post means Barcelona now have an unassailable 11-point lead over fifth-placed Betis with three games of the LaLiga season remaining.
Earlier, Barcelona had taken the lead with a first goal since January for Ansu Fati in the 76th minute. But Barca were pegged back almost immediately when former Camp Nou defender Marc Bartra headed home a corner just three minutes later.
Alba's late goal means Xavi Hernandez has achieved his main aim after taking over midseason. It also gives Barcelona a healthy five point cushion in second place, although third-placed Sevilla have played a game fewer.
"It's a weight off our shoulders," Xavi said afterward. "It could have been an even tougher season but the team has competed well since we arrived in November.
"We have not always played excellently, but we have a winning spirit and we are competitive. We've been all the other top half teams in the second half of the season. It's clear we have to pay better. We have to analyse things. But the effort has been huge since November. We're in the Champions League, a competition we deserve to be in, now we want to finish second."
Betis forward Juanmi had the first sight of goal in an entertaining match, but his effort could only find the side-netting in the seventh minute.
The Portuguese substitute, however, produced an excellent display, starting with a superb save from Ronald Araujo's bullet header minutes after entering the pitch.
Betis created more opportunities in a lively first half, but substitute Fati put the visitors ahead after the break with a low shot towards the left corner in the 76th minute.
"Ansu has a gift, others find it more difficult," Xavi said. "It's a great feeling to have him back. The goal was something we had talked about this week regarding the Betis defensive line.
"He's a difference-maker. He just needs half a chance to score, even with a bad shot. He told me best to play him wide. I told him he could change with Auba, but he wanted to be on the left. He's always a danger coming inside. He gets two chances, scores one and the other is dangerous. He has a gift. He's comfortable around the box."
Despite Bartra levelling for the Copa del Rey champions with a header, Alba sealed the game in the dying minutes.
Barcelona, who have won 10 of their last 12 league matches, are close to wrapping up second place and snatching the 3.4 million euros that the club would make by going to the Spanish Super Cup in Saudi Arabia.
Both Copa del Rey finalists -- winners Betis and runners-up Valencia -- will be joined in the 2023 Spanish Super Cup finals by top two teams in the LaLiga standings.
Betis, meanwhile, missed an opportunity to move level on points with fourth-placed Atletico in their bid to secure a spot in Champions League next season, as they remain fifth with 58 points.
Before the match, Barcelona gave a guard of honour to the Betis players for their cup victory last month.
Information from ESPN's Sam Marsden and Reuters was used in this report.