Louis van Gaal's first game in his third spell in charge ended in a 1-1 draw in Norway on Thursday (AEST), but the Oranje produced a much improved display at Philips Stadion.
Depay fired Netherlands into a deserved first-half lead from the penalty spot and doubled his tally just after the hour mark with a clinical finish.
Georginio Wijnaldum added a third and Cody Gakpo scored his first international goal as Netherlands picked up a win that keeps it level with Norway and one point behind Group G leader Turkey, which it faces next.
Montenegro entered the game level on points with its opponent and felt it should have had a penalty when Adam Marusic went down under pressure from Frenkie de Jong.
Netherlands was otherwise on top and went close through a Davy Klaassen header that hit the crossbar before their breakthrough arrived seven minutes before half-time.
Dusan Lagator dragged down Depay in the box and the Barcelona forward blasted the resulting penalty towards the top-right corner, sending Matija Sarkic the wrong way.
Milutin Osmajic wasted a promising opportunity for Montenegro early in the second half and Stefan Mugosa blazed over after pouncing on a terrible Tyrell Malacia back-pass.
Those misses proved costly as Depay sent a zipping low strike past Sarkic at his near post after being played in by fellow forward Gakpo.
It was a familiar story eight minutes later as Wijnaldum collected Steven Berghuis's pass and calmly converted just moments after Mugosa scuffed a shot at Justin Bijlow.
The goal of the match belonged to PSV youngster Gakpo, who curled a glorious shot out of Sarkic's reach from outside the box, capping a wonderful night on Van Gaal's latest homecoming.