Leclerc was forced to defend from Russell at the start with the Mercedes attacking to the outside of the Rettifilo, the Ferrari driver staying ahead.
Lando Norris had a dreadful getaway and slipped backwards, swamped by Pierre Gasly and Verstappen into the first corner.
Verstappen was stunning in the opening stages, picking off his rivals one by one to reach third by the start of the second lap.
That became second on lap five after breezing past Russell's Mercedes into the first chicane.
Sainz was on a charge from the off and having started 18th, the Spaniard had reached sixth by lap 11, scything through the midfield on medium tyres.
But a sad ending to Sebastian Vettel's final Monza outing triggered a change in strategy at Ferrari up front.
Ferrari take tactical gamble
Vettel's Aston Martin began to puff smoke at Ascari on lap 11 and ground to a halt on the run down to the same corner on the following lap.
The recovery mission for the marshals triggered a virtual safety car and allowed Ferrari to take a pit stop.
Leclerc was switched to mediums, such was the concern over the pace of the hard tyres, but left the pole-sitter a tall ask to reach the end of the race.
That released Verstappen and Russell, both remaining on their soft tyres, to pursue a more conventional pit window.
The Dutchman made the switch to mediums 13 laps after Leclerc, rejoining 10 second behind his rival before pumping in fastest lap after fastest lap to pressurise the Ferrari pit wall.
The Scuderia was forced into a second stop with Leclerc, softs fitted to the F1-75 in order to try and chase Verstappen for the race victory.
But the championship leader's pace was just too strong and the result means Verstappen can mathematically clinch the championship next time out at Singapore.
Mercedes strategy helps Hamilton, hurts Russell
Russell had struggled to keep tabs with the front two in the first stint and when pitting on lap 23, Mercedes switched the Briton to hard tyres.
The move proved to take Russell entirely out of the fight for the top two and threatened his podium with a hard-charging Sainz closing in on soft tyres.
Team-mate Hamilton, meanwhile, had made steady progress from 19th on the grid and only reached an ailing Fernando Alonso on lap 27.
But the extended stint allowed a switch to the soft tyres for the end of the race and a vintage double-overtake on Lando Norris and Pierre Gasly at the first corner helped promote Hamilton to fifth.
Ricciardo triggers late drama - but safety car spoils tension
Alonso would retire leaving Norris sixth and the recovering Sergio Perez - who dealt with an early pit stop and fiery brakes before charging to seventh.
Daniel Ricciardo spent much of the race as a rolling roadblock for Gasly, who could not make his way past the McLaren much to his frustration.
But Ricciardo pulled off in between the two Lesmos to allow Gasly through to eighth and triggering a late safety car on lap 48.
George Russell and Carlos Sainz used the opportunity to pit for fresh tyres, as did the front two drivers on the next tour around.
But with Ricciardo's car stuck on the side of the road and the safety car picking up Russell - and therefore having to release the cars to pick up Verstappen - the race was forced to end under the safety car - the opposite scenario to the controversial ending to the season last season.
A chorus of boos greeted the procession on the final lap, though on this occasion the rulebook was followed precisely.
Russell would keep third ahead of Sainz and Hamilton.
Perez finished sixth to complete a top six for the big three teams. Norris took seventh ahead of Gasly.
Nyck de Vries' job interview at Williams could hardly have been better, securing ninth on his debut after stepping in for Alex Albon late on Saturday.
The Dutchman complained of sore shoulders at the end of the race, but his superb efforts won him the driver of the day, as voted for by fans.
Zhou Guanyu rounded out the top 10 to secure a point for Alfa Romeo, ahead of Esteban Ocon who failed to rescue Alpine's weekend.
The Canadian finished behind the second AlphaTauri of Yuki Tsunoda and ahead of Kevin Magnussen, who finished last of those still running and was taken out of contention with an early penalty for leaving the track and gaining an advantage.
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