The 25-year-old Queensland right-hander was named as a replacement for injured Mitchell Marsh with in-form Australia already holding a 2-0 lead in the series.
Burns, who is yet to play Test cricket, made a career-best first-class score of 183 against New South Wales last month.
He was notified of his inclusion by chief selector Rod Marsh and Burns admitted: “It took a fair while to sink in.
“I’m over the moon at the news as it is the best Christmas present I could ever hope for.”
Team coach Darren Lehmann said Burns offered flexibility with his ability to bat anywhere in the top six.
Lehmann said: “He’s a very good player against fast bowling and he’s had a good couple of summers at the Gabba with Queensland.
“There have been some good contenders for that spot and it’s always a tough call on other players.
“He’s a very aggressive player, which is the way we like to play our cricket. We went with Joe, he’s a younger player and we feel he has something about him.”
Lehmann said selectors went with an extra batsman to replace Marsh because Shane Watson bowled well in India’s second innings and therefore could adequately fill Marsh’s role as an all-rounder.
Marsh, who injured his right hamstring while bowling on the opening day in Brisbane, will stay with the national squad and be assessed ahead of a possible return for the fourth Test in Sydney, starting on January 6.
Lehmann also said opener David Warner has a bruised left thumb after being struck by India???s Umesh Yadav on the final day of the Gabba Test.
He went on: “Last night it was just bruised so hopefully he pulls up alright and we’ll get to Melbourne and sum it up as we go.
“The initial signs are good and I’m pretty confident he’s going to be fine.”
Australia???s 13-man squad: David Warner, Chris Rogers, Shane Watson, Steve Smith (c), Shaun Marsh, Joe Burns, Brad Haddin, Mitchell Johnson, Ryan Harris, Peter Siddle, Mitchell Starc, Nathan Lyon, Josh Hazlewood