While Vettel had to settle for second place behind Bottas, it was Kimi Raikkonen who came under pressure from Lewis Hamilton during the early stages.
The Finn looked on course for a straightforward win until a rare slip saw him damage a tire.
Following the pit stop, Vettel was able to gradually close the gap to Bottas, but the Finn was too quick to be caught by the championship leader.
Valtteri Bottas says that the Russian Grand Prix win on Sunday has given him confidence to finish at the top step of the podium more often. There is no extra aggravation which we had previous year.
"This odd opportunity came over the winter to join this team and I have to thank the team for making this possible", he added after meeting Russian president Vladimir Putin before spraying champagne on the podium. "You never know in life what's going to happen and it was a great opportunity".
"It took quite a while, more than 80 races for me, but it's definitely worth the wait".
"Obviously Felipe did him a favour twice".
"I'm happy for Valtteri" he said. "Lewis was one of the first ones to congratulate Valtteri on his first race victory".
The Finn won his first grand prix on Sunday in Russian Federation and is now in third place on 63 points, 10 behind Lewis Hamilton, his Mercedes team-mate. Later, he added: "Why is my vehicle overheating, guys? It's not important", Hamilton said of the championship situation. To make matters worse, his Renault team-mate Nico Hulkenberg finished eighth to secure his second points-finish on the trot.
With the European season proper starting in Spain next week, and teams promising upgrades and a faster pace of development, the two frontrunners return to their factories with two wins apiece. He drove better than the rest of us so he deserved to win.
Having tasted success, Bottas is eager for more but acutely aware that it will be easier said than done. The Perth-born 27-year-old did his best to do that and was the fastest home outside the two powerhouses, albeit nearly two seconds off the pace.
The three-time champion was briefly pushing on the rear of third-place Raikkonen, but was ordered to back off by his team as the car's temperatures rocketed, leaving Hamilton frustrated.
Ricciardo told his team over the radio that "my right rear brake's on fire".
A crash further back in the field between Romain Grosjean in the Haas and Renault's Jolyon Palmer brought the safety auto onto the track shortly after Bottas had taken the lead.
It took 81 races to achieve, plus a little indication of his first pole at the last race in Bahrain.
Fernando Alonso continued to struggle with his McLaren-Honda race vehicle as the driver reported loss of power during the formation lap itself.