The 22-year-old put in a sterling effort to get on the podium in a stacked field in the Netherlands – the only Brit to do so on the third day of racing action.
Charlie Tanfield – who won team pursuit gold on Thursday night – finished fourth in the individual pursuit while Elinor Barker came home sixth in the omnium.
But Stewart was Britain's star of the night as he took two laps on the field – one early and one late – to finish on 49 points, behind home rider Jan Willen van Schip and Australia's Cameron Meyer – who celebrated a fifth world points race title.
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"That was everything a points race should be and everything I expected it to be," he said.
It's my first medal at worlds and to do it in the points race – it's my favourite event and I absolutely love it so I'm really happy.
"I've had enough bad days and made enough bad decisions that I knew 160 laps is a long time and even when you get to 80 laps there's still 20 minutes of racing left to go – that's a ten-mile time trial, there's still so much damage you can do to yourself.
"I never underestimated how much that early lap took out of me and I saw myself dropping down the leaderboard to seventh.
"But I knew there was a long way to go and I knew I could still do a fair bit of damage – as I proved in the last 40 laps."
Stewart finished second in just three of the 16 sprints but his two laps on the field were enough to see him onto the podium ahead of esteemed riders such as Kenny de Ketele – last year's silver medallist.
Later in the night Tanfield went for the bronze medal in the individual pursuit but just came up short.
The 21-year-old has come into these championships as an independent rider, not part of the British Cycling set up, but as a winner of the event in the last round of the World Cup he had proved himself on the global stage.
And while he gave it all he had, a second medal in his debut World Championships just eluded him.
It was a tough day for Barker in the omnium, too. The 2017 world points race champion was a late call up for the omnium after defending champion, and British teammate, Katie Archibald pulled out due to illness.
Barker finished seventh in the scratch race before winning the tempo race. But she was caught up in a crash during the elimination race and that seemed to knock the wind from her sails – eventually finishing seventh to leave her third heading into the final points race.
And a single point in the final race was perhaps an indicator of the crash taking its toll as the Netherlands' Kirsten Wild took her second gold medal of the week, in front of a vocal home crowd.
In the day's morning session Jack Carlin and Ryan Owens qualified for Saturday's quarter-finals in the men's sprint, while Dan Bigham finished ninth in the individual pursuit.
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