Karnataka first dismissed the home team, who added 94 runs to their overnight score of 342 for five, 37 minutes before lunch. The visitors then rode on some steady top order batting to finish the day just 261 runs behind the 40-time champions' total on a slow Wankhede Stadium track, where the odd ball had started to turn.
At stumps, opener Ravikumar Samarth was unbeaten on 85 having hit eight fours in his 188-ball knock, along with No. 4 batsman Manish Pandey, who had faced 15 balls and was yet to score.
Earlier, Mumbai's young batsmen Siddesh Lad and Nikhil Patil completed well-deserved individual hundreds before departing after scoring an identical 106, having commenced the day on 94 and 85, respectively.
Lad, who was dropped twice yesterday, hit 20 fours in his 135-ball knock, was dismissed by rival skipper R Vinay Kumar when he edged the pacer to wicketkeeper C M Gautam. It provided the visitors the eagerly awaited breakthrough in the morning after the sixth wicket duo had rescued Mumbai with a superb stand of 190 in 272 balls.
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Mumbai's last two wickets added a further 32 runs before their innings ended close to lunch.
For Karnataka, captain Vinay Kumar was the most successful bowler with a haul of 4 for 88, while Patel (3 for 96), Aravind (2 for 91) and leg spinner Shreyas Gopal (1 for 91) were the other wicket takers.
Brief scores:
Mumbai 1st inngs: 436 all out in 117.1 overs (Siddesh Lad 106, Nikhil Patil 106; R Vinay Kumar 4/88, Udit Patel 3/96).