The Turkish champions secured three valuable points while maintaining a clean sheet, strengthening their position in the competition and boosting their goal difference.
Speaking after Tuesday's encounter, Buruk emphasised the importance of the victory to Galatasaray's European ambitions.
"It was very important to win today," he told media.
"We won 2-0 and did not concede a goal. Both getting three points and contributing to our goal difference was very important today in terms of our goal of advancing to the next round.
"We achieved this. Our goal is, of course, to go higher."
Despite his satisfaction with the result, the Galatasaray manager expressed frustration over the playing surface, insisting that poor conditions affected both teams.
"Both teams were very tired today. The playing conditions were not good. We played on a very bad pitch," he explained.
"The pitch troubled both teams from time to time and caused passing errors."
Buruk also questioned several refereeing decisions, suggesting that Stuttgart escaped stronger punishment for some challenges during the opening half.
"There were very harsh challenges, especially in the first half. Maybe there could have been more yellow cards," he added.
"It is also necessary to evaluate whether Lemina's incident could have been a red card.
"The referee was English, but when we looked at the 90 minutes, we thought that some decisions favoured our opponent."