West Indies’ Obed McCoy claimed 6-17 as India were bowled out for 138 with two balls to spare in their innings.
Streamlined bowling brought India back into the competition, with West Indies needing 10 runs in the last over.
Bhuvneshwar, known for his death-over mastery, still had two overs left but Rohit instead handed the ball to Avesh.
Avesh, who made his India debut earlier this year, started with a no-ball Odeon Smith Took it a single.
Devon Thomas, off the next ball, hit a free-hit, a six and a four as West Indies leveled the five-match series with four balls to spare.
Rohit said that he wants to test India’s bowling depth keeping in mind the Twenty20 World Cup in Australia.
“We know what ‘Bhuvi’ does for us, he has been doing it for years. Unless you give chances to people like Avesh and Arshdeep (Singh), you will never know.
“He has the skill and talent and it’s all about backing him. I am really proud of the bowlers and the team.”
bowling coach Paras Mhambre Hopefully the series will help India identify the bowling attack positioned in Australia.
“It’s part of our preparation for the World Cup,” Mhambrey told reporters.
“We are looking at the various options that we have and are giving young people the opportunity to find out what works for us.”