Webster produced finishes of 102, 94 and 76 in a 4-3 defeat of Antonio Alcinas but Burnett suffered a loss by the same score-line against Carlos Rodriguez.
The pair then sealed a 4-2 success in their doubles match to set up a semi-final clash against tournament favourites England.
“We’re relieved to get through,” admitted Webster, after sealing a place in the last four for a third consecutive time.
“It was a bit of a scrappy tie but we got the job done and that’s all that matters.
“Yesterday it was a bit like this in the afternoon but we kicked on in the evening so hopefully that’ll be the same tonight.”
Webster grabbed a break of throw in the opening leg of his singles game, hitting double top after 14 darts.
Alcinas responded by hitting the first maximum of the match as he broke back immediately.
He then edged ahead with a fine bull finish but Webster hit back with a brilliant checkout of 102.
The Spaniard pegged double twelve to move within a leg of victory before ‘Spider’ forced a deciding leg with a clinical finish of 94.
And the left hander stepped in to seal the win with another tidy finish, this time taking out 76 after Alcinas missed four match darts.
Burnett produced a sensational 156 finish in his singles match but was edged out 4-3 by Rodriguez, leaving the result to be settled in the pairs.
Spain moved in front when Rodriguez took out 96 but Burnett levelled on double five.
Alcinas took out 70 as Spain wrestled back the lead but Wales restored parity once again when Webster followed a Burnett maximum by pinning double 14.
Webster pegged double ten for a timely break of throw before sealing the victory for Wales on double 16.
Team Wales will face the English pair of Phil Taylor and Adrian Lewis with a place in the final at stake.
By Chris Murphy