six games on court Suzanne Lenglen.
Ivanovic, who has never made it past the last 16 here since her triumph, next faces Russian ninth seed Ekaterina Makarova.
Makarova, a semi-finalist at the US and Australian Opens, was never in trouble as she beat compatriot Elena Vesnina 6-2 6-4.
In her first two matches, Ivanovic lost the first set against Kazakh Yeroslava Shvedova and Japan’s Misaki Doi. There was no repeat on Friday when the 27-year-old dismissed the up-and-coming Vekic in ruthless fashion.
“After the first two matches, I really started to get a little bit of feeling and groove, and yesterday I had easy day. Today I really had strict game plan. I had an idea what I wanted to do, and obviously the confidence the from first two matches helped me in today’s performance,” Ivanovic told a news conference.
“When you know you are working in the right direction, it always gets you results. Today I did some things better than I did in my first two matches.”
Since 2008, the inconsistent Ivanovic had reached the last 16 in Paris on two occasions, falling at the first or second hurdle twice.
The Serb has not been beyond the fourth round since she won the title in 2008 and will have to beat ninth seed Ekaterina Makarova if she is to improve on that record.
Meanwhile, Roger Federer continued his smooth progress on Friday with a straightforward 6-4, 6-3, 6-2 defeat of Bosnian Damir Dzumhur to reach the fourth round.
Swiss Federer has won all nine sets he has played at Roland Garros this year as he attempts to win the title for a second time and extend his grand slam haul to 18.
Dzumhur, 10 years younger than Federer, tested his idol on occasion, showing a nice line in drop shots, but he never really threatened to detain the second seed for too long.
Federer broke serve in the fifth game, which was sufficient to bag the first set.
A majestic backhand pass gave Federer a break at the start of the second, but then he became a little sloppy and was broken when serving at 5-2. He hit back immediately to move two sets ahead and rolled through the third with ease.