Electing to bat after winning the toss, Mumbai suffered an early setback when comeback man V Yo Mahesh got Akhil Herwadkar in his first over.
The opener was trapped leg-before for a duck.
Shaw and Shreyas Iyer, who has been picked in the India T20 team to face New Zealand, were in an aggressive frame of mind and attacked the Tamil Nadu bowlers.
In a quick start despite the loss of Herwadkar, the 50 came in the eighth over.
He raised his half-century at run-a-ball.
Shaw appeared subdued but played some splendid shots too as the Tamil Nadu bowlers struggled to contain the free-stroking Mumbai batsmen.
Iyer fell at 106 after a fluent knock of 57 which included seven fours and two sixers.
He edged a delivery from Vijay Shankar to keeper N Jagadeesan.
The Mumbai batsmen continued to look for runs and Surya Kumar Yadav added 63 runs with Shaw in just over 11 overs to keep the momentum.
The young right-hander hit 17 fours and two sixes before falling for 123 to R Ashwin.
The countrys premier off-spinner had previously got rid of the aggressive Siddesh Lad.
Mumbai skipper Aditya Tare scored an important half-century to further the teams cause and added 63 runs with the experienced Abhishek Nayar, before Vijay Shankar provided TN the much-needed breakthrough.
The all-rounder provided another wicket for Tamil Nadu by sending back Vijay Gohil in the final over of the day as Mumbai finished at 314 for 7.
Left-arm spinner Rahil S Shah went for 82 runs in his 25 overs while young all-rounder M S Washington Sundar bowled only four overs.
Brief scores (Day 1): Mumbai 314 for 7 in 89.1 overs (Prithvi Shaw 123, Shreyas Iyer 57, Aditya Tare 53, Vijay Shankar 3 for 37, R Ashwin 2 for 77) vs Tamil Nadu.
At Vizianagaram: Andhra Pradesh 278 for 2 in 89.5 overs (G Hanuma Vihari 143 batting, D B Prashanth Kumar 127) vs Odisha.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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