Hamilton took the checkered flag on lap 66, becoming the first driver to score points in 30 straight Grands Prix, and further raising the bar for most consecutive race starts, with 213.
"I'm OK", he responded on the team's radio link.
"Our tires didn't last as long as for the others, so we couldn't follow the same strategy", Vettel said.
Red Bull's Max Verstappen celebrates on the podium with his trophy after finishing in third place.
This marks Verstappen's first podium of the 2018 season. Race by race we are understanding the tyres more.
"In many ways, this is a weekend in which nothing went right", team principal Maurizio Arrivabene said.
The race leader had to navigate multiple virtual safety auto periods during the race, as well as a full safety vehicle deployed for an opening lap crash caused by Romain Grosjean who had spun in Turn 3.
As you were: Mercedes took its first pole position since the season-opener in Melbourne.
Vettel had pitted when the virtual safety auto was deployed to slow the field, something that had played in his favour at previous races.
Spirited display from Ireland in first test match
Ireland also had a chance to become the third team, after England and India, to win a Test after following on. Babar, who was dropped on 9, was run out 20 runs short of the target.
Ferrari, who had been very strong in qualifying for the past three races, could not quite find the improvement they needed in Q3 and Vettel did well to be within one-tenth of Hamilton.
But Ferrari's misjudgement saw Max Verstappen take advantage - despite an earlier coming together with Lance Stroll (him again) resulting in a virtual safety vehicle - to take his first podium of 2018 for Red Bull.
But Red Bull can ill afford a repeat between the highly competitive pair, having insisted they apologise to the entire team after their Baku collision cost them valuable points.
The season-opening Australian Grand Prix aside, Mercedes has been chasing Ferrari in the early part of the season, and even with Hamilton's victory in Azerbaijan, questions were still being asked about their performance levels.
Hamilton, who also holds the record of 74 pole positions, had started from the top slot for only the second time this year. Starting from P14, Leclerc benefited from the eliminations of Grosjean, Hulkenberg and Gasly to push up into the top ten before maintaining a solid driving ability throughout the race. But because of Botas' late devastation, along with Kimi Raikkonen's second-place finish, Ferrari jumped ahead of Mercedes in the constructor standings.
Sebastian Vettel has defended the pit stop that put paid to his hopes of victory at the Spanish Grand Prix on Sunday, insisting Ferrari had "no option".
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