Nathan Ake scored twice in a game that saw the Netherlands have 50 attempts without allowing one at the other end, as Koeman's team got up and running in Euro 2024 qualifying following last week's heavy defeat to France.
Ake's second-half double came after Memphis Depay had put the Dutch ahead against the Group B minnows in Rotterdam on Monday.
It marked a 19th straight defeat in Euro qualifying for Gibraltar, who had Liam Walker sent off for a rash challenge on Dutch debutant Mats Wieffer.
In need of a response following their hammering in Paris, the Netherlands had control from the off, and the lead came in the 23rd minute.
Following a patient move, Denzel Dumfries – back from suspension – supplied an inviting cross that Depay glanced in.
Georginio Wijnaldum volleyed just over while Wout Weghorst twice went close – spurning a great chance when he shot wide from eight yards out before Dayle Coleing produced a smart reflex save to deny the Manchester United striker.
The Dutch doubled their lead five minutes after the break, Dumfries helping a deep cross back into the centre for Ake to head home.
Gibraltar were reduced to 10 men less than 60 seconds later. Walker caught Wieffer with a dangerous challenge, leaving referee Morten Krogh with little choice but to brandish a straight red.
Coleing denied Donyell Malen and Cody Gakpo while Roy Chipolina blocked Weghorst on the line, but Ake's deflected strike eight minutes from time added gloss to an easy victory.