In what was just her fourth match since the Australian Open, Kanepi - who missed all of February with a shoulder injury - made a sluggish start, dropping serve in the third game. However, the No.5 seed fought back impressively to break the big-serving Babos twice to nick the opening set.
The second was just as evenly contested, yet this time it was Babos who held her nerve at the business end, coming out on top in a flurry of late service breaks to force a decider. Kanepi looked the stronger player in the third but, despite twice being a break up and holding points for a 5-2 lead, she struggled to put away her teenage opponent.
Nevertheless, Kanepi was not to be denied, and in a match where big serving set the tone, it was fitting that she closed out a 75 46 76(2) win with her 12th ace.
"It was a difficult and really long match," Kanepi said. "Timea played really well and pushed me all the way. Thankfully I played a really solid tie-break, cut down the errors and came through.
"Spending two and half hours on court is tiring, but I'll be ready for my next match."
Joining Kanepi in the quarterfinals will be Mona Barthel, Petra Martic and Alizé Cornet. Barthel, the No.6 seed this week, brushed aside Alberta Brianti 63 62 for her 24th win of the season, while Martic defeated qualifier Johanna Konta, 62 63, and Cornet squeaked past Patricia Mayr-Achleitner, 62 26 75.
Last week, Jankovic played the spring's opening clay court event in Charleston and it took her a while to adjust to the low-bouncing hardcourts of the Farum Arena. But despite a few scares along the way - Robson held a set point in the second set tie-break - the Serb eventually triumphed 64 76(8) after nearly two hours.
"I found it pretty tough moving from clay to indoor hard, where the ball goes very low and fast, so I found the adjustment quite difficult," Jankovic said. "Laura played really well, but I knew I had to hang in there. I was a little nervous as it was my first match here, but I'm glad I managed to come through with the win."