Georgia Stanway gave the Lionesses a first-half lead in Crete before Maya Le Tissier and Alessia Russo put England firmly in control with two more goals after the break.
Kallisti Brouksair pulled one back for the hosts, giving Greece a glimmer of hope ahead of Tuesday's return leg in Leicester against the 2023 World Cup runners-up.
"We started well, they started very aggressively too, but we stayed calm. That's what I'm the most pleased with. I think we have a couple of things to work on too," England manager Sarina Wiegman told ITV.
"At moments in the game we weren't tight enough, the ball wasn't moving quick enough... I think if we would have created these opportunities we had, we could have scored so many more goals."
Two-time reigning European champions England will face Ukraine or Slovakia if they qualify for the second play-off round, which will determine who qualifies for next year's World Cup in Brazil.
Ukraine won the first leg of their tie 3-1 in Slovakia.
The Republic of Ireland thrashed Kazakhstan 8-0 in Astana, while Scotland won 2-0 away to the Czech Republic.
Wales earned a 1-0 victory in Albania, with 1995 champions Norway winning 2-0 away to Romania.
Italy took a 2-0 lead over Belarus as the Netherlands demolished Hungary 8-0. Sweden, Portugal and Switzerland all recorded comfortable first-leg wins, while Belgium edged Poland 1-0 as both teams missed a penalty.
The winners of the 16 first-round ties advance to the second round scheduled for late November and early December.