World Cup winners France battered Moldova 4-1 while Cristiano Ronaldo's Portugal were held to a 0-0 draw by Ukraine.
World champions France started their quest for Euro 2020 glory with a comfortable 4-1 victory over Moldova in their opening qualifier in Chisinau.
Antoine Griezmann, Raphael Varane and Olivier Giroud all scored in quick succession before the break, emphasising the difference between the Group H teams.
Les Bleus finished second in their Nations League group, missing out on progressing to the finals after defeat to Holland, but there was no danger of another surprise defeat once they took a 23rd-minute lead.
Paul Pogba's exquisite ball over the top was volleyed home from close range by Griezmann, who turned provider three minutes later as his corner was headed home at the near post by Varane.
Giroud then moved to joint third on France's all-time, alongside David Trezeguet on 34 goals, after coolly volleying beyond Moldova goalkeeper Alexei Koselev from Blaise Matuidi's cross in the 36th minute.
Kylian Mbappe got on the scoresheet three minutes from time after being teed up by substitute Thomas Lemar before Vladimir Ambros grabbed a late consolation for Moldova.
Elsewhere in the group, Burak Yilmaz and Hakan Calhanoglu scored the goals as Turkey won 2-0 in Albania while Iceland beat Andorra by the same scoreline thanks to Birkir Bjarnason and substitute Vidar Kjartansson.
Portugal - who alongside England have reached the Nations League semi-finals in June - drew a blank in a goalless draw against Ukraine in Lisbon.
Also in Group B, Leandro Barreiro and Gerson Rodrigues were on target as Luxembourg came from a goal down to defeat Lithuania 2-1.
Raheem Sterling's hat-trick helped England begin their campaign with a flourish, Gareth Southgate's men brushing aside the Czech Republic 5-0 at Wembley.
Sterling opened the scoring in the 24th minute after getting a boot to Jadon Sancho's teasing cross and then won a penalty on the stroke of half-time, which Harry Kane converted.
The Manchester City forward scored twice more just after the hour before Tomas Kalas put the ball into his own net to complete a miserable night for the Czechs.
Group A rivals Bulgaria and Montenegro drew 1-1 in Sofia, Todor Nedelev's penalty eight minutes from time ensuring a stalemate after Stefan Mugosa had opened the scoring just after half-time.
Euro 2020 qualifying results
- Bulgaria 1-1 Montenegro
- Albania 0-2 Turkey
- Andorra 0-2 Iceland
- England 5-0 Czech Republic
- Luxembourg 2-1 Lithuania
- Moldova 1-4 France
- Portugal 0-0 Ukraine