Spain had broken the deadlock in the 53rd minute through Esther Gonzalez before Ella Toone forced extra-time with a volley six minutes from the end of regulation time.
Georgia Stanway’s rocket into the back of the net five minutes into extra time would decide the tie as England advanced to their fourth major tournament quarter-final in a row.
“It is what we came here to do, we didn’t pack up, we left all our stuff there, we weren’t going home today,” the centre-back said.
“And I think you just saw the fight. A convincing win is good but a comeback like that for a team, it does a lot for your confidence and the emotion we finished that game with was unbelievable.
“That’s the difference, you go from group stage to a knockout game and I never felt like we were going to lose, it was just a matter of time and the formation change helped.
“I have seen her [Georgia Stanway] do that all her career so I wasn’t surprised so to speak, more relieved that she dis it so early into extra time. As soon as it left her foot I could see it.”
England were put under real pressure for the first time in their home tournament having breezed through the group stages without conceding.
But it was Spain who would get first blood as substitute Athenea Del Castillo passed the ball to Gonzalez who rifled home to send shockwaves through the Brighton crowd.
Substitutes and Manchester United teammates Alessia Russo and Toone combined to move England ahead, Russo heading down for Toone to volley home and keep England’s hopes of a first major title alive.
Williamson added: “It is important to have a test and you expose yourself to that and you can prove to yourself that you can come through the other side that is what is important about what we just did.
“Regardless of the opposition that is the first test we have had in the tournament, and it is important we played with our heart.”
Sarina Wiegman’s tactical substitutions made reinvigorated England as her Spanish counterpart Jorge Vilda’s failed, but it was starter Stanway who had the final say.
The Spain defenders sat off a marauding Stanway who took aim from the edge of the box, with Sandra Panos helpless in goal as England moved ahead for the first time in the game.
Wiegman was back on the bench having missed England’s final group game as she isolated with Covid and celebrated jubilantly as the full-time whistle blew and England advanced to face either Sweden or Belgium in Sheffield next Tuesday.
The England manager said: “We didn’t expect an easy game. I am so incredibly happy this was just a crazy day and crazy game and a game of such a high level I think everyone who watched the game was entertained and the crowd was incredible.
“It could have been both sides because both sides wanted to win and played well. Spain are very good in a possession game so that was hard for us to press the ball, stay compact,
“But we had moments in the game where we were good on the ball, where we created chances in front of the goal. It is a really big commercial for women’s football.”
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