The striker netted a close-range finish and a penalty in a five-minute burst before the break at Riazor, to join an elite band of players in reaching the quarter-century goal mark for La Roja.
Yeremi Pino's effort and two goals from Pablo Sarabia in the second half made it a handsome victory for Luis Enrique's hosts, who looked a class above their frequently overwhelmed visitors.
With just two shots to their name throughout the entire match, Arnar Vidarsson's side seldom troubled their hosts, whose performance offered a timely reminder of their Qatar 2022 credentials eight months out from the World Cup.
Having struggled to carve Albania open until the last quarter-hour in its friendly on Sunday (AEDT), Spain initially looked in similar trouble over the opening 30 minutes in A Coruna once again.
But with over 80 per cent of the ball to their name, it felt like only a matter of time before they found the opener, and Morata duly provided nine minutes out from the interval.
The Juventus striker latched onto Hugo Guillamon's pass, stepped over by Carlos Soler in a superb feint, and smuggled home a finish at the left post past Runar Alex Runarsson.
Morata was on hand to double Spain's lead three minutes later when Dani Olmo was fouled in the box by Birkir Bjarnason, sending the goalkeeper the wrong way with a cool finish from the spot.
Iceland's hopes of a fightback looked particularly remote at merely two goals down, but they became nonexistent when Pino nudged home at the left post moments after the interval.
Sarabia's subsequent header around the hour mark, followed by a simple tap-in from the same player, added gloss to the scoreline in a supreme Spanish performance.