Bumrah struck early to remove Hashim Amla and Quinton de Kock.
Yuzvendra Chahal was the pick of the Indian bowlers, with figures of 4-51 that came after Bumrah's early assault.
Kohli said Bumrah would be a "massive factor" for his team at the World Cup.
"The way he's bowling right now, it's so good to see because you see batsmen literally clueless against him and he's rushing people," said Kohli.
"To be honest, I haven't seen Hashim get out like that in one-day cricket... I think it's a great thing and it's a testimony to his hard work.
"He understands this is his time and he's making the most of it so hats off, he's brilliant. He's going to be a massive factor for us here."
Bumrah got Amla to edge a rising delivery to Rohit Sharma at second slip. He then struck again in his next over, with De Kock (10) edging to Kohli at third slip.
"Well, one thing you know with facing Jasprit is you have to play good cricketing shots against him and you have to back your technique against him," said Kohli.
"Because if you are hesitant a little bit, he senses that and he's all over you."
Chahal soon struck twice with his leg spin including South African captain Faf du Plessis for 38.
Kuldeep Yadav chipped in with 1-46.
"I think Kuldeep bowled really well, he controlled the game from one end and I think Chahal was outstanding," said Kohli of the spin twins.
"Even with a little bit of turn and help from the pitch, what he was able to bring out of the wicket was truly brilliant to see. He has mental belief in his strength and full credit goes to him."