Hamilton started on pole at Jeddah, with a crash in qualifying by Verstappen giving the Mercedes driver the chance to potentially take the lead in the standings with one race to go.
However, the safety car was brought out after Mick Schumacher spun into the wall between turns 21 and 22, with Hamilton choosing to dive into the pits on lap 10 to switch to hard tyres as Verstappen stayed out and took track position.
With race officials wanting to repair damage to the tyre wall, the red flag was then waved, forcing all drivers to return to the paddock and giving Verstappen the chance to switch his tyres to the harder compound without using a pit stop.
Hamilton expressed his annoyance on team radio, saying: "Why is there a red flag? The tyre wall looks fine. I know the reason for the red flag."
He continued: "Have they said what the reason was? The tyre wall that looks fine. [Chief strategist] James [Vowles], that was a huge gamble we took."
Vowles replied: "It was a risk we knew, but we didn't think it [the red flag] would happen."
However, there was then another twist from the subsequent standing start, from which Hamilton got a much better getaway.
He looked to have the lead going into the first corner, but Verstappen went off track and swooped ahead of him, with Esteban Ocon also getting ahead of the Briton.
Verstappen was not allowed to keep the lead, with a tense radio negotiation between Red Bull and race director Michael Masi resulting in another standing restart with Ocon on pole, Hamilton second and Verstappen third.