Hamilton was catching the German on every lap but ran out of time and finished nearly seven seconds behind.
It is his second win this season, having taken the flag in Australia and his 44th career victory.
"Over the years, nothing changes and I feel pain in my heart, ' Hamilton said". To make matters worse he was then slapped with a five-second timed penalty for driving too slowly into the pits. But a slow stop for the Finn forced Hamilton to back off to buy himself some time, in the process baulking Red Bull's Ricciardo for which the triple champion collected a five place time penalty.
Bottas was first told to move aside for Hamilton on lap 25 as Vettel built a six-second lead and again with 10 laps remaining.
"The beginning for us to be that close was great".
McLaren endured a desperately disappointing day with Stoffel Vandoorne failing to start and Fernando Alonso, who expressed his anger at his sluggish Honda engine throughout the race, retired in the closing stages.
The four-time world champion, endured a torrid time a year ago, and was even warned by the Ferrari hierarchy that he would have to earn his next contract - with his current deal expiring at the end of this season.
Hamilton managed to recover from that setback as he got past Ricciardo and Bottas, but could not catch up to the race leader in time.
Ricciardo's Red Bull team-mate 19-year-old Max Verstappen crashed out of the action after a brakes failure.
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- For the first time ever in his F1 career, Fernando Alonso has retired in all the three first Global Positioning System of a season.
Running in third, Hamilton then urged his Mercedes team to let him past Bottas with Vettel sailing off into the distance.
Things weren't quit as smooth for Mercedes as Hamilton had to serve a five-second penalty for being too slow in the pit lane, and Bottas struggling with his vehicle all race long - failing to add a maiden victory to his first career pole the previous day.
He apologised to the team afterwards for his actions, which he said might have cost him victory.
Mercedes also suffered during the pit stops, with a loss of power in the garage preventing the team from using its wheel guns properly. Lewis Hamilton (Mercedes) +00:06.600; 3.
Valtteri Bottas who was previously at Williams, has yet to triumph.
Kimi Räikkönen came home 4th in the other Ferrari. Felipe Massa (Williams) 00:54.326; 7. Romain Grosjean (Haas) 01:14.865; 9.
Hulkenberg scored Renault's first points of the season. Force India are the only team outside the top two to have had both cars scoring in every race. Pascal Wehrlein (Sauber) 1 lap; 12.
Stroll's lap 12 collision with Sainz, who was blamed by stewards and handed a three place grid penalty for the Russian grand prix, brought out the safety vehicle.
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