Victor Wembanyama scored 17 points, grabbed 14 rebounds and blocked six shots to help the Spurs wrap up a 4-1 win in their best-of-seven Western Conference first-round series.
The Spurs, who clawed back from a 19-point deficit to win game four, burst out of the blocks and pushed their lead to as many as 28 before taking a 65-45 lead into half-time.
Portland unleashed an 11-0 run that cut a 20-point fourth-quarter deficit to nine, but the Spurs responded.
De'Aaron Fox scored 13 of his 21 points in the final period and Wembanyama - playing his second game after missing one with a concussion - delivered two massive blocks as San Antonio won a playoff series for the first time since 2017.
"It did feel like we let our foot off the gas," Spurs coach Mitch Johnson said. "NBA games are long, it's not an excuse. That means we've got to be better, but zero lead changes is zero lead changes.
"We have a lot of things we can be better at," Johnson added of his young team, who had six players score in double figures. "But we're ready to step up to the plate and swing."
Seeded second in the West, they are chasing a sixth NBA title and their first since 2014.
They will face either the Denver Nuggets or Minnesota Timberwolves in the Western Conference semi-finals.
The Timberwolves lead the Nuggets 3-2 in their series.