The Welsh left-hander held off a fightback from his Dutch opponent to reach his first major semi-final for almost four years.
He followed up wins over Stephen Bunting and Adrian Lewis by stamping his authority on the quarter-final tie, dropping just one leg as he raced into a two set lead.
Klaasen then found his feet and replied in kind to take the match to a deciding set, where Webster hit a crucial 86 finish on the bull to move 2-1 ahead.
Klaasen took the next to force a last-leg shoot-out, which Webster won on double top.
“I played solid and I’m really pleased,” said Webster, who will take on Michael van Gerwen on Friday. “At 2-0 up I started missing my double in, but I can’t be too critical of myself because I dug in.
“This is the beauty of the PDC – opportunities are always there for someone and thankfully I’m taking mine at the minute. I feel really good and I’ve worked hard, so bring on the semi-finals.”