There was a late consolation for the home side courtesy of Sam Vokes's header in the final minute, but they could have no complaints at the final score on a night which saw them outclassed by Spain, who have made it three wins from three under Enrique.
Wales were without record goalscorer Gareth Bale due to a groin injury, and it quickly became evident that they would miss the Real Madrid man when Spain took control of possession early on.
It took just eight minutes for the visitors to turn that possession into an opening goal when Alcacer continued his recent good form at club level by firing his finish into the roof of the net after Wayne Hennessey's punch had only gone as far as Jose Gaya.
Wales boss Ryan Giggs may have already been fearing the worst at that stage as Spain took complete control, although he would have expected more from his defenders for the second goal as they made it far too easy for the visitors to double their advantage.
Indeed, the defence went AWOL to leave a queue of Spanish players unmarked from a free kick, and captain Ramos gleefully accepted the gift by nodding Dani Ceballos's delivery past Hennessey and into the bottom corner - the defender's fifth goal of the season and 15th for Spain.
The hosts were struggling to muster anything in response, with the closest they came in the opening 30 minutes being when Harry Wilson looked to catch David de Gea out with a long-range free kick which flew over the crossbar.
Spain, by contrast, had the game effectively wrapped up before the half-hour mark when Alcacer - top scorer in the Bundesliga this season - doubled his tally for the night, latching on to Wilson's botched clearance to drill a volley past Hennessey from close range.
It was the first time that Wales had conceded three goals in the first half of a home game for 12 years, and they would have been relieved that it did not get any worse before half time as Spain took their foot off the pedal.
There was an attacking moment of note for the home side before the interval, though, as Wilson's corner was met by Ethan Ampadu, although De Gea watched the looping header all the way and allowed it to clip the outside of the post on its way behind.
Both sides made changes at the break, but little changed in the pattern of the match and Spain created a big chance to add to their advantage just three minutes after the restart when Hennessey thwarted Alvaro Morata, who had been played through from a quick free kick which caught Wales sleeping.
Spain were in complete control at this stage as they began to pass the life out of Wales, and Alcacer squandered a chance for his hat-trick at the end of one such move when he nodded his effort over the crossbar.
The visitors came even closer moments later when Suso cut inside and curled an effort towards the top corner which crashed against the top of the crossbar with Hennessey beaten.
Wales finally drew a save from a Spain keeper when Aaron Ramsey's long-range strike on the turn was comfortably held by substitute Kepa Arrizabalaga, but Enrique's side quickly resumed control of proceedings and Morata nodded one effort wide when he should have done better.
Rodri was the next to go for goal - an effort which was easy for the keeper - but it took a last-ditch block from Spain debutant Jonny Castro to deny Vokes a chance at the other end moments later as Wales made a rare foray forward.
The fourth goal did not flatter Spain, though, as they finally opened their second-half scoring with just over 15 minutes remaining when Bartra sent a looping header into the far corner for his first international goal.
Wales did finally begin to improve in the final 10 minutes - with Bournemouth midfielder David Brooks standing out for the hosts - and he was involved in the buildup to their first shot on target after 86 minutes when Tom Lawrence's volley was comfortably gathered by Kepa.
Spain immediately went up the other end as Rodrigo Moreno and Jonny forced a couple of saves from Hennessey in quick succession, but Wales were looking a lot more threatening and it took a well-timed challenge from Gaya to deny Connor Roberts a late chance.
The hosts were rewarded for their late endeavour with an 89th-minute consolation, and it was no surprise to see Brooks at the heart of it as he delivered a sumptuous cross into the box with the outside of his boot which Vokes rose to meet and glance past Kepa into the bottom corner.
It proved to be the final meaningful action of the match as Spain made it three wins and 12 goals from three games under Enrique, whereas Wales have now lost successive matches for the first time under Giggs.
The Dragons will now turn their attention to Tuesday's UEFA Nations League clash with the Republic of Ireland, while Spain host England in Sevilla on Monday night.
WALES (3-4-3): Hennessey; Gunter, A Williams (Chester 46'), B Davies (Richards 62'); Roberts, Allen (Smith 62'), Ampadu (King 50'), John (Lawrence 62'); Wilson (Brooks 46'), Vokes, Ramsey
SPAIN (4-3-3): De Gea (Kepa 46'); Azpilicueta (Jonny 63'), Ramos (Bartra 46'), Albiol, Gaya; Saul (Koke 46'), Rodri, Ceballos; Suso (Rodrigo 81'), Morata, Alcacer (Aspas 73')