Real Madrid midfielder Isco then shot high into the net in the 23rd to complete an equally impressive move of short floor passes between David Silva and Koke, while Rodrigo's cousin Thiago Alcantara scored the third three minutes later with a thumping header from debutant Alvaro Odriozola's cross.
"We are feeling very happy because we've achieved an objective that was really important for Spanish football," said Spain coach Julen Lopetegui.
"It was very tough but we have done it while playing brilliant football."
The only downside for Spain was a shaky performance from Gerard Pique, who was booed by large sections of the Alicante crowd every time he touched the ball following his support for last Sunday's referendum on Catalan independence and was substituted by Lopetegui after an hour was up.
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Despite missing influential players such as Alvaro Morata, Andres Iniesta and Dani Carvajal through injury, the 2010 world champions looked as irresistible as in their 3-0 win over Italy last month, scoring three times in the space of 11 minutes.
They set the tone with a sweeping team move early on which saw Rodrigo miss the target from a Jordi Alba cross, and the Valencia forward wasted another chance for an early goal when he headed a perfectly weighted cross from Isco wide.
Rodrigo, who was recalled to the Spain squad for the first time in three years after spearheading Valencia's impressive start to the Liga campaign, put those misses behind him with a clinical finish after being played in by Isco.
It was the start of a rampant spell from Spain which effectively put the game to bed and killed off Albania's faint hopes of a playoff place as they fell five points behind Italy, who are second in the group.
Lopetegui's side lowered their intensity after the break but still could have stretched their lead as Isco saw a free kick tipped over the crossbar and substitutes Marco Asensio and Saul Niguez each came close to getting on the scoresheet.
Spain have taken 25 points and scored a remarkable 35 goals from their nine Group G games, only dropping points in their 1-1 draw away to Italy last year, and travel to Israel for their final qualifier on Monday with their place in Russia assured.
"Spain are one of the favourites to win the World Cup because their players have so much experience and today we saw their quality," said Albania coach Christian Panucci.
"The best thing about this team is the commitment, the passion and the enthusiasm shown by every player that has been with us, from those who played a lot to those who were only called up once," added Lopetegui.
"Without a doubt the attitude is the best thing about this team and if we continue to show the same hunger and humility we will be a great team."
(Reporting by Richard Martin; Editing by Rex Gowar)
