Eta, whose appointment triggered derogatory social media comments, took over as interim manager in April following the dismissal of Steffen Baumgart and will lead the side until the end of the campaign before moving to the women’s team next season.
After back-to-back defeats in her first two games in charge, Union picked up a point against Koln earlier this month and Eta told broadcaster DAZN she was delighted to get a win at the fourth attempt.
“It's been about doing the job as well as possible, getting points, winning games, and that's what we've worked for each day," she said.
“You're happy, you want to win games. That's always the case, and so it was today as well. The way it happened was great too, how we managed to pull it off.
"We have invested a lot. So it's great that we could get the three points here today, and show a good performance," Eta said.
"That's important for me, too."
Union face Augsburg on Saturday in their final match of the season.