Tare, the 25-year-old right-handed wicketkeeper-batsman, hit 31 fours and a six during his 417-balls knock. He spent 369 minutes at the crease.
This was the first time Tare crossed the three-figure mark in first-class cricket.
Tare's teammate, Rohit Sharma scored 166 off 254 balls and spent just 109 minutes in the middle. The batsman's knock was decorated with 16 fours and a six.
The two shared 278 runs for the third wicket.
Resuming on 306 for two, Tare and Sharma were rarely troubled by the Saurashtra bowlers after the two started from their overnight scores of 122 and 72, respectively.
Mumbai tightened the grip further, taking the wickets of Bhushan Chauhan and Sagar Jogiyani to leave Saurashtra struggling at 42 for two at stumps.
At the end of second day's play, the hosts trailed Mumbai by 564 runs with with eight wickets remaining in the first innings.
Shitanshu Kotak (6) and Jaydev Shah (28) were at the crease when stumps were drawn.
Earlier, even after the massive stand between Tare and Sharma was broken, there was no respite for Saurashtra as Abhishek Nayar went on to score an aggressive 69 off 84 balls with the help of six fours and a six.
Saurashtra will now have to bat of their skin to save the follow-on.
Mumbai 1st innings: 606/5 declared in 162 overs (Aditya Tare 222, Rohit Sharma 166)
Saurashtra 1st innings: 42/2 in 15 overs.