England - represented by Luke Humphries and Luke Littler - started quickly against Spain but fought through a small blip to seal an 8-5 victory on Saturday.
The top seed raced into a 3-0 lead but saw the Spaniards win three legs on the bounce themselves before Littler eventually finished them off with a checkout on 89.
“I’m proud of the way we fought back. It wasn’t the worst performance in the world, but it wasn’t our best," Humphries said.
“Just get through that first game, and then tomorrow it feels like we can relax a little bit and play how we know we can.”
England will face Wales, who also won by an 8-5 scoreline after defeating the USA. Jonny Clayton and Nick Kenny enjoyed a solid performance on Saturday that included two 141 checkouts across the match.
Also into the quarter-finals is reigning champions Northern Ireland following their narrow 8-7 win against Belgium.
Josh Rock and Daryl Gurney were in danger of crashing out at one point, finding themselves down five sets to two after a slow start, but they regrouped and sealed a superb comeback victory.
"At no stage did we think we were beaten,” Gurney admitted.
“Until the very last throw, we still believed we could win that game, and we weren’t in front until we won it.
“That tells you we’re champions. We’re here to win; we’re trying to go back-to-back, but now this game is over, we start afresh tomorrow!”
They take on Latvia in the last eight, while Scotland - who demolished Norway 8-0 - will face Ireland. The Irish booked the spot in the quarters with an 8-5 triumph over Poland.
World Cup of Darts round of 16 results: