Alvaro Morata scored twice for Spain, while Fabio Quagliarella entered the record books for Italy in Tuesday's European Qualifiers.
Morata ended a 16-month wait for an international goal when he broke the deadlock against Malta with a neat angled finish in the 31st minute.
The striker on loan at Atletico Madrid from Chelsea then met a Jesus Navas cross in the 73rd minute to seal a 2-0 win.
Spain's comfortable victory - in the absence of Luis Enrique, who missed the match for family reasons, saw them go top of Group F, with maximum points from their opening two matches.
Chelsea goalkeeper Kepa Arrizabalaga came in for Manchester United's David de Gea, while there were starts for Valencia defender Jose Luis Gaya, Atleti midfielder Saul Niguez and Barcelona's Sergi Roberto.
Italy posted a second successive win at the start of their qualifying campaign by brushing aside Liechtenstein 6-0 in Parma.
Quagliarella became Italy's oldest international goalscorer at 36 when he converted the first of two penalties (35, 45+3) for Roberto Mancini's side in the Group J clash.
Liechtenstein played the second half with 10 men after Daniel Kaufmann was sent off for conceding the second spot-kick, but Italy had already inflicted too much damage as Quagliarella's successful conversion made it 4-0.
Stefano Sensi (17) and Marco Verratti (32) had set the Azzurri on their way, while Juventus star Moise Kean added a fifth 20 minutes from time and debutant Leonardo Pavoletti completed the scoring (76).
Over in Group D, Conor Hourihane's first senior international goal handed the Republic of Ireland a narrow but deserved 1-0 over Georgia as Mick McCarthy was treated to the perfect homecoming.
The Republic, who had scraped their way to a 1-0 win over minnows Gibraltar on Saturday evening, were more than value for the points as they won by the same score at the Aviva Stadium with the Georgians rarely looking like ending their wait for a win over the Irish at the 10th attempt.
Hourihane's 39th-minute free-kick - moments after tennis ball protests against outgoing Football Association of Ireland chief executive John Delaney - ultimately settled the game, although goalkeeper Darren Randolph had to make a good reaction save to deny Valerian Gvilia an equaliser and Jaba Kankava hit the post five minutes from time.
Denmark launched a stunning late comeback against Switzerland to draw their opening qualifying match for Euro 2020.
Remo Freuler, Arsenal midfielder Granit Xhaka and Breel Embolo had given the Swiss a 3-0 lead heading into the closing 15 minutes.
Huddersfield defender Mathias Jorgensen's 84th minute header appeared little more than a consolation but Christian Gytkjaer set up a tense finale before Brentford's Henrik Dalsgaard took advantage of panic in the Swiss defence to secure a remarkable point deep into stoppage time.
There was more late drama in Group F's Scandinavian derby as Norway and Sweden shared another 3-3 thriller in Oslo.
AZ Alkmaar forward Bjorn Johnsen gave the home side the lead from close range in the 41st minute and Bournemouth striker Joshua King headed in a second on the hour.
Sweden got a lifeline with 20 minutes left when Viktor Claesson slotted home after Andreas Granqvist's penalty had been saved.
Robin Quaison then equalised with five minutes left, his shot deflecting in off Norway defender Havard Nordtveit.
There was, though, still time for a frantic finish as Quaison struck again in stoppage time - only for Norway to earn a share of the points when Ola Kamara headed home in the seventh minute of added time.
Greece scored twice late on to come back and draw 2-2 away at Bosnia and Herzegovina.
Edin Visca and a free-kick from Juventus midfielder Miralem Pjanic inside the opening 15 minutes saw the hosts ahead in Zenica.
However, the Greeks started the fightback through a 64th-minute penalty from Konstantinos Fortounis before Pjanic was shown a straight red card for a foul on Jose Zeca.
Substitute Dimitris Kolovos secured a dramatic draw five minutes from time, heading the ball in off the underside of the crossbar.
Earlier on Tuesday, Finland had got their Group J campaign up and running with a 2-0 win away to Armenia, who remain bottom. Fredrik Jensen and Pyry Soiri scored the goals.
Romania defeated the Faroe Islands 4-1 in Cluj, where Ludogorets forward Claudiu Keseru scored twice. Ciprian Deac and George Puscas scored Romania's other goals, with Viljormur Davidsen's penalty the Faroes' only reply.