With the victory, Portland, the 16th seed, moves on to the quarterfinals to face the No. 8 seeded San Antonio Spurs Saturday afternoon. Portland shot.283 (12 for 41) through intermission.
That's when Layman, who struggled in the first half, scored 10 points in the final frame to help the Blazers pull away.
Portland will play Memphis in Sunday's semifinals.
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Derrick White came off the bench to lead the Spurs with 20 points. Layman led the Blazers with 23 points. The Blazers knocked down 13 of 22 shots in the second half. Toronto's Fred VanVleet hit a three with 14 secodns to play, though Johnson and Swanigan would make their free throws late to secure the upset.
The 6-9 Layman was 8 for 18 from the field, though only 1 for 6 from 3-point range, in his 35 minutes.
Rookie big man Caleb Swanigan finished with 13 points, 11 rebounds and two steals in the win over the Raptors, recording his third double-double in Las Vegas.
Guard R.J. Hunter also scored 16 points and dished five assists. Tipoff is scheduled for 1 pm on ESPN2.