However Spain's joy was overshadowed by injuries suffered by David Silva and Alvaro Morata, while Gerard Pique was once again jeered by the home support on Friday.
Victory took Vicente del Bosque's men five points clear of third-placed Ukraine, who Spain travel to face in their final qualifier in Kiev on Monday.
"We've managed to qualify at last. We knew we couldn't be complacent, we played very well and managed to get the win," Cazorla told Television Espanola.
Due to an injury to Sergio Ramos, the hosts started without a Real Madrid player in the line-up for the first time ever in a European qualifier.
However, it was the treatment once again dished out to Pique for mocking Madrid during Barcelona's treble-winning celebrations last season that caused controversy.
Pique's teammates and Del Bosque appeared dismayed that their calls for the centre-back to be supported had fallen on deaf ears for the second consecutive home fixture.
"The truth is it is uncomfortable," added Cazorla.
There was further worry for Del Bosque when Silva was forced off with a right ankle sprain after just 10 minutes that could also prove bad news for Premier League leaders Manchester City.
Spain's movement in the final third was much better than in previous qualifiers and twice they passed up huge chances to go in front as Jordi Alba blazed over from point-blank range before Cazorla smashed a shot against the crossbar.
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