Badani's patient 84 came off 221 balls while Shrivastava faced 129 deliveries during his impressive knock.
Badani stitched two crucial partnerships -- first 97 runs with Shrivastava and then 53 with Amol Jungade for the ninth wicket -- to take Vidarbha past the 200-run mark after the visitors' lost their top three batsmen early.
Sent into bat, Vidarbha's Faiz Fazal, Aniruddha Chore and former India opener Shiv Sundar Das failed to give their team a good start as they slumped to 16 for three.
But Shrivastava and Badani steadied Vidarbha's sinking ship with a vital 97-run fourth-wicket stand.
Shrivastava's knock came to an end when he was cleaned up by Firdaus Bhaja. He crafted his innings with 11 boundaries.
After Shrivastava's dismissal, experienced Badani held his ground but wickets kept tumbling at the other end until he found an able partner in No. 10 Jangade, who ended the day unbeaten on 27 that came off 57 balls.
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Badani decorated his unconquered knock with 12 hits to the fence.
For Baroda, medium pacer Bhaja picked up three wickets for 36 runs, while Murtuja Vahora (2/85) and skipper Yusuf Pathan (2/35) scalped two wickets apiece.
Brief Scores:
Vidarbha 1st innings: 219 for eight in 88 overs (Hemang Badani 85 not out, Shalabh Shrivastava 63; Firdaus Bhaja 3/36).