Bangladesh declared on 560-6 after Mushfiqur Rahim passed 200, with Zimbabwe then losing two wickets to leave them in deep trouble.
The spectacular Motera Stadium was opened by Donald Trump and Narendra Modi in front of an enthusiastic crowd.
India captain Virat Kohli rued his side's first-innings batting display in the loss to New Zealand.
New Zealand wrapped up the first Test inside four days against the ICC's top-ranked team.
South Africa levelled the Twenty20 series against Australia at 1-1 thanks to some excellent bowling in the closing overs.
Mominul Haque finished day two unbeaten on 79 as Bangladesh trimmed Zimbabwe's first-innings lead to just 25.
New Zealand's bowlers left India facing the prospect of a likely defeat in Wellington.
Sri Lanka took a 1-0 lead in the three-match ODI series with West Indies thanks to heroics from Wanindu Hasaranga.
Nayeem Hasan was the star for Bangladesh, who were frustrated by Craig Ervine's century for Zimbabwe.
New Zealand are in a promising position against India, and Tim Southee picked out the moment he felt his side stole the advantage.