Siraj took 4-40 off 14 overs and fellow paceman Bumrah picked up 3-42 as the visitors were shot out in 44.1 overs in Ahmedabad on Thursday.
West Indies No.7 Justin Greaves top-scored with 32 for the tourists.Â
Left-arm wrist-spinner Kuldeep Yadav picked 2-25 in 6.2 overs.
By stumps, India trailed by just 41 runs. Lokesh Rahul was unbeaten on 53 and skipper Shubman Gill was 18 not out.
After choosing to bat, West Indies lost three wickets in the first 10 overs and five wickets by lunch as the top order looked to play shots.
Siraj grabbed 3-19 in the morning, but Bumrah took a couple of overs to find his rhythm.
West Indies opener Tagenarine Chanderpaul was caught behind for an 11-ball duck.
Bumrah struck in the seventh over to remove the other opener, John Campbell, caught behind for eight.
Captain Roston Chase and Shai Hope added 48 for the fifth wicket, until Yadav bowled Hope for 26.
After lunch, Chase was caught behind off Siraj for 24.
Greaves and debutant Khary Pierre (11 off 34) helped their side pass 150, before Pierre was trapped lbw by Washington Sundar.
India's innings began after tea, and Rahul put on 68 off 110 balls with Yashasvi Jaiswal (36 off 54).
Sai Sudarshan, playing at his Indian Premier League home ground, was lbw for just seven as the home side slipped to 2-90.
Play was extended by 22 minutes because of rain, and after the resumption Rahul reached his half-century off 101 balls, despite suffering leg cramps.
West Indies have not beaten India in their past 25 Test encounters - home or away - dating back to 2002.
