The British number five's only previous experience of grand slam tennis came in the first round of Wimbledon this year when, as a wild card, she pushed top-30 player Christina McHale all the way.
Konta marked her first time in qualifying by winning three matches against higher-ranked players last week and on Tuesday she produced the best performance of the lot to defeat world number 59 Babos 6-2 7-5.
The scoreline only told half the story, with Konta, who will climb considerably from her current ranking of 203, winning the final six games of the match and saving 10 set points in a remarkable ninth game of the second set.
In the opening set she had been much the better player against an out-of-sorts Babos and then showed her fighting spirit to win in an hour and 48 minutes in stifling conditions.
Konta, who has Hungarian parents, already looks a very valuable addition to the British ranks having switched allegiance in May from Australia, the land of her birth, to the country where she has lived for the last seven years.
Her victory made it three British players into the second round after Monday's wins for Andy Murray and Laura Robson, while Heather Watson and Anne Keothavong both lost to top-10 players.