Depay scored twice, including one from the penalty spot while squandering another spot-kick, to take his goal tally in the World Cup qualifiers to nine -- more than any other player from a European team.
Virgil van Dijk opened the scoring with a header from a Depay corner after nine minutes and there were also second-half goals for Denzil Dumfries, Arnaut Danjuma and Donyell Malen.
The result left the Dutch two points ahead of Norway in the Group G standings with two matches left to play. The Netherlands compete for their qualifiers next month away against Montenegro and home to Norway.
Depay had a 19th-minute penalty saved by Gibraltar debutant Bradley Banda but then made up for it with a tap-in from close range two minutes later before netting his second penalty effort on the stroke of halftime.
It was a fourth win in a row for veteran coach Louis van Gaal, who was brought back to help ensure World Cup qualification after Frank de Boer resigned following the team's Euro 2020 failure.