In reply, Baroda made 63 for 1 in 26 overs in their first innings in the Group C match at the Wankhede Stadium.
It was a combination of impressive swing and seam bowling from right-arm Atit (5 for 50) and left-arm Meriwala (5 for 52), aided by poor shots attempted by the Mumbai batsmen that sent the home team to the doom.
Inserted in, Mumbai plunged deep into trouble to be 5 for 2 by the third over, with Atit sending back in-form opener Prithvi Shaw and Ajinkya Rahane for ducks, and never fully recovered from these early setbacks.
Shaw was bowled through the gate, while Rahane edged a drive to rival captain Deepak Hooda at first slip.
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Tare was fifth out just before lunch, taken with the hosts and 41-time champions tottering at 103 for five. The wicketkeeper-batsman stayed at the wicket for 143 minutes, played diligently in contrast to some of his teammates when facing 82 balls and struck 8 fours.
Among the batsmen who disappointed and was out before lunch was Shreyas Iyer, fresh after playing in the T20 International series against New Zealand, who made 28.
He was dropped by wicketkeeper Mitesh Patel on 5 and then nearly edged to the slip cordon on 22 before chasing a wide ball moving further away to be caught at second slip.
He used the short ball to good effect to send back Shardul Thakur and Gohil.
Baroda, who began their reply after tea, played out 11 overs and advanced to 25 before they lost opener Nur Ahmed Pathan for 14, caught by Tare behind the stumps off left arm pacer Roystan Dias.
Solanki played some attacking shots to remain unbeaten on 32, with six fours, along with a sedate Aditya Waghmode (15) as Baroda were 108 runs behind Mumbai going into day two tomorrow.
Brief scores: Mumbai 171 all out (Aditya Tare 50; Atit Sheth 5 for 50, Lukman Meriwala 5 for 52). Baroda 63 for 1 (Aditya Waghmode 15 not out, Vishnu Solanki 32; Roystan Dias 1 for 15).