Stoke manager Mark Hughes saw his side take a 10th-minute lead against his former club at Ewood Park when Peter Crouch volleyed home at a corner, only for King to head in an equaliser in the 36th minute.
After Geoff Cameron was sent off for holding King back, Rudy Gestede put Blackburn in front with a penalty on the stroke of half-time before King completed his treble with quick-fire goals early in the second period.
Earlier, Brown Ideye continued his fine recent form by scoring twice as West Bromwich Albion beat a woeful West Ham United 4-0 at The Hawthorns to also reach the last eight.
The Nigeria striker, West Brom’s £10 million ($15.2 million, 13.4 million euros) record signing, scored either side of a 25-yard strike from James Morrison to take his tally to four goals in three games.
West Ham substitute Morgan Amalfitano was sent off after shoving Chris Brunt in the face and Saido Berahino added a fourth to rubber-stamp West Brom’s progress to the quarter-finals for the first time since 2008.
“It was a good performance and we’re very pleased, but there’s still a long way to go in the league and we need that bit of luck with the next draw,” West Brom manager Tony Pulis told BT Sport.
“Brown Ideye was fantastic. He is scoring goals and that takes the weight off your shoulders, and he looks a good player.”
Five-time winners West Brom have now won five of the eight matches since Alan Irvine was succeeded as manager by Pulis, who is bidding to emulate his achievement of guiding Stoke City to the final in 2011.
Blackburn and West Brom were joined in the quarter-finals by Reading, who won 2-1 at fellow Championship side Derby County courtesy of a late strike by former Portsmouth, Middlesbrough and Everton striker Yakubu.
Liverpool tackle Crystal Palace later on Saturday, when Brendan Rodgers’s side will attempt to arrest a run of six games without victory at Selhurst Park.
Holders Arsenal host Middlesbrough on Sunday, while third-tier giant-killers Bradford City, conquerors of Premier League leaders Chelsea in round four, take on Sunderland at Valley Parade.
Manchester United face a third consecutive tie against lower-league opposition on Monday when they visit third-division Preston North End.