The German - like Hamilton, chasing a fifth world championship - saw the fast-finishing Bottas devour his lead in the closing laps but held on to win by 0.7 seconds despite badly deteriorated tyres.
Hamilton, who started ninth after being given a five-place penalty for a gearbox change, said he was satisfied.
In one thrilling maneuver during the race Hamilton overtook three drivers in one go.
No driver has won the opening two races and then failed to win the title since Keke Rosberg in 1982.
"With 10 laps to go, I came on the radio and said I had everything under control", Vettel said.
He had earlier told his team that everything was under control but that, he recognised later with a smile, was not the reality.
"When they told me the pace of Valtteri at that time, I thought no way I could do that".
"I tried to keep it as clean as possible". It worked but only just. Vettel stopped for the soft rubber on lap 19 with Raikkonen in a lap later for the same tyres. He was able to successfully hold on to the lead into the first corner, but his Ferrari team mate Kimi Raikkonen lost out to Valtteri Bottas.
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Ferrari mechanic Francesco Cigarini suffered a double leg fracture when he was run over by Kimi Raikkonen during Sunday's Bahrain Grand Prix.
Hamilton could take some consolation in equalling Raikkonen's all-time record of 27 successive points finishes.
"I don't believe in all those things", he said before turning toward the present. Despite half his Mercedes and Red Bull competition starting the race well adrift of Vettel, his foes knifed their way through the field with Verstappen and Hamilton making up multiple positions in the first few laps. He was left with a rear left puncture and limped back to the pits for new tyres before retiring the vehicle due to damage from the early incident.
"I was doing the maths, oh, that pace, he's going to catch me".
"But I think there was still enough space on the left". But he sustained too much damage after puncturing his left rear tire.
Behind the battle of the big teams, Pierre Gasly was the "best of the rest" in fourth place at the wheel of a Toro Rosso-Honda (yes, a Honda powered car).
Vettel started on the front row with Raikkonen, and Bottas started third. But they finished a lap down.
Sauber's Marcus Ericsson banked his first points since the 2015 Italian Grand Prix in ninth place after running an extremely long first stint on softs and switching to mediums to execute a one-stop strategy to good effect.