With the tie level at 2-2 from the first leg, the hosts took an early lead on Sunday in Valencia through Mikel Oyarzabal's penalty.
The Dutch equalised through a spot kick from Memphis Depay early in the second half, only for Oyarzabal to put Spain back ahead.
Aston Villa's Ian Maatsen scored a debut goal with 11 minutes left to make it 2-2 on the night.
Lamine Yamal and Dutch substitute Xavi Simons, from a penalty, both scored in extra time, sending the tie to a shootout.
After Noa Lang struck the crossbar with the Netherlands' fourth kick, Yamal's effort was also saved.
But Spain goalkeeper Unai Simon saved from Donyell Malen in sudden death with Pedri then converting to give Spain victory.
France also needed penalties to beat Croatia, coming back from a 2-0 first leg deficit.
Michael Olise's free kick early in the second half and Ousmane Dembele's late goal made it 2-0 to Les Blues at Stade de France.
France goalkeeper Mike Maignan saved two spot kicks before Dayot Upamecano slotted in to secure a 5-4 success.
Germany survived a second-half comeback from Italy to advance 5-4 on aggregate after finishing 3-3 in Dortmund.
Defending a 2-1 lead from the first leg at the San Siro, Germany looked in control after captain Joshua Kimmich slotted in a penalty and Jamal Musial added another.
Tim Kleindienst made it 3-0 just before halftime, but Italy fought back with two goals from Moise Kean and then an added-time penalty from Giacomo Raspadori.
Germany will host Portugal in June after Cristiano Ronaldo's side came through 5-2 after extra time against Denmark in Lisbon.
Ronaldo had an early penalty saved, but later scored his 136th international goal in the second half.
Christian Eriksen's late goal looked to be sending the Danes through, but Francisco Trincao struck with four minutes left to force extra time and then added another early in the first period before substitute Goncalo Ramos wrapped things up.
In the second leg of the play-offs, Serbia maintained their place in the top tier on the back of a 2-0 win over Austria in Belgrade, closing out a 3-1 aggregate victory.
Greece secured promotion to League A after an impressive 3-0 win over Scotland at Hampden Park to overturn a 1-0 deficit.
Belgium avoided dropping out of League A after they beat Ukraine 3-0 in Genk, having lost the first leg 3-1.
Turkey were promoted to League A after a 3-0 win in Hungary completed a 6-1 aggregate victory.
