Mirza and Bhupathi fought back from a break down in the opening set to trounce the Polish-Mexican partnership after Jans-Ignacik smacked a backhand wide on match point.
Seventh seeds Mirza and Bhupathi, who added the clay title to the Australian Open crown they won in 2009, were left bemused at the end when they were handed the wrong silverware by mistake.
A giggling Mirza passed on the rectangular runners-up tray to Gozalez before the red-faced French Open official handed the right trophy to the Indian duo.
"I want to thank my partner who is one of the best doubles players in the world and I'm lucky to play with him," Mirza told the sparse crowd on Philippe Chatrier Court.
"Last year I was a runner-up in the women's doubles so I'm delighted to finally get a Roland Garros trophy.
"It is my partner's birthday today so now I don't have to think of a gift to give him," Mirza added as the fans burst into an impromptu rendition of 'Happy Birthday'.
Bhupathi added: "It's been a while since I won a grand slam and this is the first time I've won a grand slam trophy as a father so this is for my four-month old daughter back home."