The sought-after Fiji forward rejected several offers from clubs across Asia, including the most lucrative contract ever offered by the Phoenix, according to Stuff.
Krishna's decision to join the two-time ISL champions came down to his ancestral ties to India, the country from which his "forefathers" originated.
"It was not just about me, it was about my family," Krishna said of the move.
"My wife, [Naziah Ali], is coming with me. She's more familiar with the Indian culture and that was also one of the reasons why I chose to go there.
"Obviously my forefathers came from there 140 years ago so I'm going back after 140 years. I'm excited, I'm happy and I know I've made the right decision."
Krishna's short-term move to the Kolkata-based club follows the likes of Robbie Keane, Helder Postiga and Luis Garcia to have worn the red and white stripes.
Though just 31 years of age and fresh off an incredible A-League campaign as the Johnny Warren medallist and Golden Boot winner, Krishna's chance to come "full circle" proved an opportunity too good to pass up.
"It comes down to who values me more and, being in the A-League, it was just hard," he said.
"We rejected ATK at the start but they never gave up hope. They never lost hope on my potential and that was the key, they valued me more than any other.
"At the same time, I look forward to introducing the club and its fans to Fiji, my homeland. I'm eager to promote Fijian tourism, people and culture while in India.
"It's disappointing to leave the Phoenix but that's part of life and we all have to move on. I just want to wish the club and Ufuk [Talay] all the best for the coming season.
"The clubs kept coming and I wanted to stay in the A-League but it comes down to who values me more and then I talked to my family and I think ATK was the right club. They valued me more and that's all I can say."