Verstappen held off Ferrari's Lewis Hamilton to win by 2.50 seconds, with Charles Leclerc third, as the McLarens of Lando Norris and Oscar Piastri touched and spun on the final lap when fourth and fifth.
A heavy shower 10 minutes before the scheduled start meant a delay of 30 minutes.
All cars began on intermediate tyres with some sectors of the Marina Bay Street Circuit much dryer than others as the field embarked on two formation laps behind the safety car.
The eventual standing start saw Russell start on the front row alongside pole-sitter Verstappen, and it was the Englishman who got away better.
His lead was short-lived as he lost control after touching a treacherous white line on the final bend of the opening lap.
He slid sideways into the pit lane wall, bringing out a safety car.
It was another setback in a season of bad luck for Russell, who will start from the back of the grid in Sunday's Grand Prix because of engine penalties.
"It was a miscalculation," Russell admitted to Sky Sports.
"Came through the corner, had more battery deployment than I thought, pushed me a little bit wider than I hoped and went on to the white line. Silly mistake. Big consequences."
It left Hamilton, who had a brilliant start from sixth on the grid, in second place having leapfrogged Ferrari team-mate Leclerc and the two McLarens.
The big beneficiary of Norris and Piastri's "papaya spat" on the last lap was championship leader Kimi Antonelli, who inherited fourth place in his Mercedes having started seventh on the grid.
Fifth was RB's Liam Lawson, sixth was the Haas of Oliver Bearman, seventh was Audi's Nico Hulkenberg and Norris got going again to limp home in eighth and round out the points scorers.