Services failed to take even a single wicket in the day as Agarkar (113 batting) and Tare (108 batting) added 211 runs for seventh wicket.
It was a patient innings from the Mumbai duo after resuming the day's play from a precarious 199 for six.
Aside from Sachin Tendulkar (56) and Abhishek Nayar (70) Mumbai's famed top order had failed to impressive on the first day.
Tare and Agarkar frustrated the Services bowlers all day, leaving a lot of deliveries.
The absence of a quality spinner came back to bite the hosts, who looked very pedestrian on a low, slow wicket.
While Tare faced 324 balls, his captain faced 237 balls in 294 minutes, hitting 12 fours.
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Tare got his second century of the season as well as his first-class career. The wicketkeeper batsman had scored 222 against Saurashtra at Rajkot earlier this season.
Meanwhile, at Rajkot, Sheldon Jackson (107) hit his third first-class century as Saurashtra went on to post a big first innings total of 477.
Jackson and Kamlesh Makvana (85) added 106 runs for the sixth wicket as Punjab bowlers failed to remove the lower batting order.
Vishal Joshi (55 not out) and Makvana added another 91 runs for the seventh wicket before the Punjab bowlers got their act together and removed the final three batsmen without much trouble.
Siddharth Kaul and Sarabjit Ladda took three wickets apiece while former India international Harbhajan Singh picked up two.
Punjab in reply, were 41 for no loss when stumps was called.
Brief Scores:
At Rajkot: Saurashtra 477 all out in 155 overs (Sheldon Jackson 107, Kamlesh Makvana 85, Shitanshu Kotak 54, Jaydev Shah 87; Siddharth Kaul 3/119, Harbhajan Singh 2/102, Sarabjit Ladda 3/109) vs Punjab 41 for no loss in 17 overs.
At Delhi: Mumbai 380 for six in 143 overs (Sachin Tendulkar 56, Abhishek Nayar 70, Ajit Agarkar 113 batting, Aditya Tare 108 batting; Suraj Yadav 2/49, Shadab Nazar 2/104) vs Services