The hosts, after being restricted for a modest first innings total of 261, hit back strongly to bundle out their Maharashtra state rivals for a paltry 113 in just 41.3 overs and then rattled up 179 for one in the second essay to hold a commanding lead of 327 overall with two days left in the game.
Tare, who opened instead of first innings centurion Wasim Jaffer, was unbeaten with a power-packed 110 in which he carted South Africa-bound pacer Umesh Yadav for two successive sixes, the second of which helped him reach his hundred. In all, Tare hit 13 fours and three sixes while facing 114 balls.
Another South Africa-bound batsman Ajinkya Rahane made up for his first innings failure by remaining unconquered on 62.
The two batsmen added 133 runs in just 26 overs to demoralise the Vidarbha attack which sorely missed their best bowler Sandeep Singh who limped off the field after suffering a foot injury when trying to stop a ball.
In the morning, Vidarbha wrapped up the Mumbai first innings quickly after the hosts resumed at overnight 254 for 8.
Jaffer was last out for 139, after adding six runs.
Vidarbha replied in a sorry fashion and were tottering at 72 for 8 against the hosts' three-pronged pace attack of Zaheer Khan, Shardul Thakur and Akbar Khan, who shared seven wickets before recovering to get past the 100 mark.
Barring opener Faiz Fazal (22) and Rashmi Ranjan Parida (34 not out), the rest were packed off in a jiffy with Zaheer, included for the two-Test series in South Africa, bagging the wickets of Shalabh Srivastava and Hemang Badani in the same over to finish with two for 30.
Akbar Khan got two for 28 in his debut while Thakur secured three for 40.
The other three wickets fell to Abhishek Nayar, Akshay Dabholkar and part-timer Kaustubh Pawar.
Brief scores: Mumbai 261 (W Jaffer 139; Sandeep Singh 4 for 35, Ravi Jangid 3 for 48) and 179 for 1 (A Tare 110 not out, A Rahane 62 not out). Vidarbha 113 (R R Parida 34 not out; S Thakur 3/40, Zaheer 2/30, Akbar Khan 2/28).