The order, placed at the Dubai air show, is for 273 A320neos and 157 A321neos, the re-engined version of the Toulouse-based company's best-selling plane.
The final cost to Indigo Partners, which invests in low-cost airlines, is not known, since buyers usually negotiate a discount on the catalogue price, especially for large orders.
"Bravo to all the Airbus teams, who have just concluded the biggest order of their history: excellent news for France and Europe," Economy Minister Bruno Le Maier tweeted after the announcement.
Boeing clocks up order, too
It more than doubles Airbus's 2017 order book, which stood at 288 at the end of October.
For a brief moment it pushed the European company ahead of rival Boeing in numbers of planes ordered - with 718 orders compared to 605 - but later in the day the US-based company, which was also present at the Dubai air show, reported an order from Gulf airline fly Dubai for 225 medium-haul 737 MAX aircraft.
In terms of market share, however, Airbus is still ahead at 55 percent.
Both companies are offering re-engine versions of their top sellers, which should mean a 15 percent reduction of fuel consumption.
The Airbus planes ordered by Indigo Partners will go to the US's Frontier Airlines, Mexico's Volar is, Chile's Jet Smart and Hungary's Wizz.