Javier Clemente's squad will confidently leave for London after the Barcelona midfielder's 84th-minute goal clinched a 2-1 win over Romania at Elland Road - and a quarter-final meeting with England at Wembley on Saturday.
Amor's timely strike, from a diving header, after Alfonso Perez had nodded on a Sergi Barjuan cross, came a day after the 28-year-old became the proud father of a boy, Daniel.
It was his second goal for Spain and Amor, who had been on the field for 20 minutes after coming on as a substitute for Juan Antonio Pizzi, said: "Everything has come at once for me.
"I became a father and now I have scored a vital goal for my country. I feel very happy."
Spain, who finished runners-up in Group B behind France, are fast becoming late-goal specialists.
But Amor was unable to follow the example of team-mate Jose Caminero, who marked his 85th-minute equaliser against France on Saturday by raising his top to reveal a T-shirt bearing the name of his daughter.
"I couldn't do that because I wasn't wearing anything underneath my shirt," he laughed.
Coach Clemente, whose side went ahead after 11 minutes through Deportivo La Coruna striker Javier Manjarin only for Florin Raducioiu to level after 29, admitted: "Our players were very nervous and very tired.
"Romania were technically a very fine team and it was only our big hearts that saw us through after a slow start.
"I would like to congratulate Romania, even though they are out of the competition, because they have played some excellent football.
"They didn't deserve to go out because they are a better technical team than some in the Group."
For Romanian coach Anghel Iordanescu, the blow of a third Euro 96 defeat in three games was only slightly softened by the pre-match assurance that he would remain in charge for the 1998 World Cup qualification campaign.
"Our participation in this tournament was a failure, but we will not make it into a big drama or tragedy. We must learn the lessons and start again," he said.
"The plain fact is the Spanish players wanted to win more than we did. They fought to the end and they deserved their victory. I can only congratulate them.
"The first half was quite even. There was lots of goal-mouth action at both ends and it must have been a nice spectacle for the spectators.
"But in the second half we never matched Spain in the aerial battle. They forced us to retreat and eventually made their breakthrough.
"But the simple reason for our poor form is that we have faced three formidable opponents. They are all good strong teams, stronger than us, and that is why we have not gone further."
Romanian midfielder Gheorghe Hagi, who marked his 100th international appearance by supplying the precise through-ball from which Raducioiu finally broke his side's tournament goal drought, admitted to an extreme sense of disappointment.
"I am particularly sad because we did not realise our full ability in the tournament, even though we showed some good moments in every game," he said.
"All the games we played in the group were very equal and rested on who took the chances. We had some problems in defence, sure, but we missed a lot of chances."Reuse content