The airport's passenger total soared to 1,106,998 - the first time it has topped 1 million in March.
January and February are now the only months yet to reach seven figures.
Further growth is expected this summer with flights to Boston being launched next month and on 12 European routes between now and October.
March's passenger total increased by 14.8 per cent compared to 11 per cent in February.
The airport's growth over the last year also gathered pace, from 7.1 per cent in February to 8.2 per cent last month, to 14.6m passengers.
Rivals Glasgow, which no longer publishes passenger figures itself, appears to be increasingly trailing in Edinburgh's wake.
Its most recent available figures, supplied by the Civil Aviation Authority, showed passenger numbers fell by almost 6 per cent in February to 533,281 - just over half of Edinburgh's 939,412.
Glasgow's annual total also fell in February by nearly 3 per cent to 956,569.
Edinburgh said its higher March figure was in part due to the knock-on disruption of the Beast from the East snowfall on 28 February last year.
Chief executive Gordon Dewar said: “To get our first ever million passenger March is another milestone for us and shows our variety of destination and types of holidays and trips are really providing for our passengers.
“While these figures are good, we always need to look ahead and consider what is on the horizon and uncertainty caused by Brexit, the expensive aviation tax we face and the impact it has on our airlines.
"We will continue to work to mitigate and manage those things as best possible and for the benefit of our passengers.”
Among Edinburgh's most popular destinations last month were Madrid, Athens, Lisbon, Gdansk, Warsaw, the Canary Islands, Alicante and New York City.
However, Norwegian ended its budget flights to Stewart in New York state - the last of three US routes which it launched only two years ago.
Other upcoming new routes include Philadelphia, Billund and Luxembourg this month.
Flights to Bodrum, Bourgas, Corfu, Bergen, Stavanger and Guernsey will be added next month, and to Milan's two airports and Aalborg in June followed by Vienna in October.