Australia lost an early wicket before veteran batsmen David Warner (41) and Steve Smith (46) stitched a 69-run partnership.
Former captain Virat Kohli made an early impact with a diving slip catch off Bumrah to dismiss Mitchell Marsh for a duck off six balls.
The left-handed Warner scored 1,000 runs in the World Cup, surpassing the previous best record held by Indian batsman Sachin Tendulkar.
Warner needed 19 innings compared to Tendulkar’s 20 innings.
Both Warner and Smith valued their experience with a steady approach on a difficult batting pitch as India rotated their bowlers.
Kuldeep, the left-arm wrist spinner, broke down the stands to send Warner back, caught and bowled amid the roar of the crowd, which had almost filled the stadium unlike the empty seats in other matches so far.
Smith stuck with Marnus Labuschagne as runs came at a premium and the stingy Jadeja got his first wicket with his left-arm spin.