Tye, who had taken a hat-trick on IPL debut against Rising Pune Supergiant at Rajkot, struck twice in two balls and returned with figures of 4-0-12-3 to leave the hosts at 22-3 in the fifth over.
Besides Tye, Ravindra Jadeja (2/28) snapped two wickets and Basil Thampi (1/34) and Ankit Soni (1/28) picked up one wicket each as Lions dished out a disciplined bowling performance.
Bangalore, who were bundled out for 49 -- the lowest total in IPL history -- by Kolkata Knight Riders two days ago, were left tottering at 61-5 in 10 overs.
After an economical opening over by Nathu Singh, Kohli hit the first boundary for Bangalore when he flicked a Basil Thampi delivery for six over squareleg in the next over.
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Gayle too picked up a four behind point in the last ball of the next over by Nathu as RCB reached 17-0 in three overs.
With Thampi keeping the pressure in the next over, a frustrated Kohli ended up hitting straight to Aaron Finch at short fine leg as RCB lost their first wicket.
Gayle then nicked a Andrew Tye delivery straight to wicket-keeper Dinesh Karthik as RCB lost both their openers inside the powerplay.
At 22-3, Kedar Jadhav (31) joined AB de Villiers and the 32-year-old from Maharashtra blasted three successive fours off Thampi in the next over as RCB scored 35-3 in the first six overs.
Jadhav hit another four off Ravindra Jadeja and then clobbered debutant leg-spinner Ankit Soni for a six over midwicket to bring up the fifty for Bangalore.
However, Jadeja cleaned up Jadhav in the next over when the batsman dragged one on to his stumps as RCB slumped to 58 for 4 in 9 overs.
Negi (32) then picked up successive fours off Jadeja in the 11th over before blasting Soni for a couple of sixes in the 14th over to bring up the team 100.
But the Delhi allrounder ended up giving a catch to Thampi at deep midwicket next as RCB further slipped to 110 for six.
RCB lost Mandeep, Samuel Badree (3) and Sreenath Aravind (9) cheaply after that and it was only after 30 balls that RCB could manage a boundary when Aniket Choudhary (15) hit Thampi for a four in the 19th over.