The British number one struggled initially in her second round tie against the Hungarian world number 26, but won 13 of the last 16 games to eventually ease to a 7-5 6-2 victory.
Konta plays US Open finalist and number five seed Karolina Pliskova in the third round, with a place in the quarter-finals at stake.
A run deep into the draw in Beijing would boost Konta's hopes of making the WTA Finals in Singapore, and also making the top ten in the world rankings and becoming only the fourth ever British woman to do so.
Battling Konta beaten by Radwanska in final
But rather than thinking ahead, Konta was just pleased to get through her rain-delayed match against Babos when not playing her best tennis.
"I think I'm most pleased with coming through a match like that. I think both of us were struggling to find our level out there," Konta said.
"Definitely when there's been a rain delay and a day of play has been missed it does break up the rhythm a little bit.
"I think that showed in both our tennis out there. So it was just about who could compete a little bit better and I was fortunate I was able to find that little bit extra and come through that."
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