After a less than convincing performance in their 2-1 victory over Romania, Spain arrived at El Molinon hoping for a morale-boosting result on a night when Sergio Ramos made his 167th appearance and equalled Iker Casillas' caps record for the national team.
Rodrigo's goals - one in each half - and a late double from substitute Alcacer made sure the Euro 2012 winners avoided embarassment against Group F's stragglers, but the visitors deserved credit for stifling Spain's strikers for long periods.
Head coach Robert Moreno, taking charge of Spain for the second time since he was appointed on a full-time basis, will be glad to have claimed another three points but will see room for improvement ahead of the next round of matches.
Thiago Alcantara's skill and ingenuity caused the visitors problems early on and his flick into the box allowed Mikel Oyarzabal to square to Rodrigo, who stroked calmly into the empty net with goalkeeper Gunnar Nielsen stranded.
Thiago was denied a wonder-goal when his immaculate volley from the edge of the box at a corner was saved by Nielsen at full stretch, and Rodrigo sent a free header over the crossbar from point-blank range as the Faroe Islands left the field relieved to be trailing by just one goal at half-time.
The second half was four minutes old when Rodrigo broke clear down the right flank, cut into the box and unleashed a left-footed shot that took a wicked deflection off Heini Vatnsdal's thigh on its way into the net.
Spain survived a scare when Joannes Bjartalid burst forward and fired a shot that David de Gea saved with his legs, but Alcacer side-footed Thiago's cross into the net from close range to make it 3-0 with a minute left.
Borussia Dortmund frontman Alcacer, who has scored in every game he has played for club and country this season, then nodded home Jose Gaya's cross at the near post to make it 4-0 in second-half stoppage-time.
What does it mean? Spain still not quite ticking
After a nervy performance at the back in their 2-1 victory over Romania, Spain enjoyed a quieter night defensively but looked flat going forward and laboured for long spells against weak opposition.
Thiago makes the difference
Spain had enough creative talent on the pitch to blow their lowly visitors away, but only Thiago seemed in particularly good form, and his performance deserved a goal when Nielsen saved his scintillating effort.
The Faroe Islands looked terrified when Suso scampered to the by-line in the opening few minutes but that proved to be his last such foray in a poor showing from the AC Milan playmaker.
Spain's campaign resumes on October 12 when they travel to Norway, while the Faroe Islands entertain Romania on the same evening.