"We nailed the race," said Sainz after his second career win for the Italian team and first since Silverstone last year.
The Spaniard eked out his hard tyres for 42 laps, winning at the slowest possible speed, for Ferrari's first victory since Charles Leclerc triumphed in Austria in July last year.
No team has ever won every race in a season of more than 10 rounds but Red Bull had looked capable of doing so until Singapore struck.
Mercedes trimmed Red Bull's lead in the constructors' standings to 308 points, keeping them waiting at least another weekend for the title.
Verstappen, who started on the hard tyres, moved up to second with Perez fourth but they were soon overhauled and plunged down the field after stopping under normal conditions.
Persons:
Ferrari's Carlos Sainz Jr, Mercedes, Lewis Hamilton, Caroline Chia, Sainz, Red, Norris, Perez, Carlos Sainz, Max Verstappen's, McLaren's Lando Norris, Red Bull, George Russell, Hamilton, Lando, eked, Ferrari's, Charles Leclerc, Verstappen, Sergio Perez, LAWSON, Aston Martin's Fernando Alonso, Leclerc, Pierre Gasly, Oscar Piastri, Liam Lawson, Daniel Ricciardo, Kevin Magnussen, Haas, Logan Sargeant's Williams, Alpine's Esteban Ocon, Russell, Carlos, Alan Baldwin, Ken Ferris
Organizations:
Singapore, Circuit, Prix, Lewis Hamilton REUTERS, Hamilton, Ferrari, DRS, Silverstone, Alpine, McLaren, New, Formula One, Ferraris, Thomson
Locations:
Singapore, SINGAPORE, Italy, Austria, Monaco, Japan, AlphaTauri, London