Captain Sergio Ramos put the visitors ahead from a corner in the sixth minute and Jesus Navas slammed the ball into the far corner in the 19th for Spain, who were still without coach Luis Enrique, who is on extended leave for personal reasons.
The unfancied Faroes struck back on the half-hour mark though when Klaemint Olsen rose to flick a header from defender Odmar Faero into the net on the volley, although Spain restored their two-goal advantage four minutes later.
Midfielder Isco fired a shot against the post which bounced off goalkeeper Teitur Gestsson and trickled into the net to be given as an own goal, while Spain left back Jose Gaya scored the best goal of the game in the 71st minute with a classy chip.
Spain top the group with nine points from three games while Sweden are second with seven after a straightforward 3-0 win at home to Malta.
Romania are third on four points after snatching a 2-2 draw at Norway after trailing by two goals.
Striker Robin Quaison put Sweden ahead against Malta in the second minute by following up on the rebound and Viktor Claesson increased their lead early in the second half with a lofted finish, set-up by a classy backflicked pass from Marcus Berg.
Teenage forward Alexander Isak added the third goal late on after coming on as a substitute.
Norway had salvaged a point in an epic 3-3 draw in their last outing against Sweden with a last-gasp goal but they were the victims of a stoppage-time equaliser against Romania.
Tarik Elyounoussi gave the hosts the lead in the 56th minute and former Real Madrid midfielder Martin Odegaard looked to have sealed Norway's first victory of the campaign when he struck in the 70th.
Yet visiting striker Claudiu Keseru had other ideas, pulling a goal back in the 77th minute with a header from close range before completing the comeback in stoppage time with another strike.
(Reporting by Richard Martin; Editing by Ken Ferris and Toby Davis)
