Bulatovic's penalty shot gave Montenegro a 23-22 quarterfinal win over former world champion France. In its first semifinal of a major competition, Montenegro will play Spain.
"I told myself, 'I have to score, I will score,' and I did," Bulatovic said. "I am so happy and so pleased for the team."
Defending champion Norway will take on South Korea in the other semifinal on Thursday.
"It was simply a phenomenal success for us," Montenegro coach Dragan Adzic said. "The whole Olympic competition is an amazing school for us as we are still learning to be a part of major sporting events."
In a close match, France went 13-11 ahead after 25 minutes, the first time a team led by more than one goal difference. France later led 17-14, before Montenegro started to dominate the game.
Despite missing three penalty shots, Montenegro levelled at 22-22 before Bulatovic found the net for her sixth goal. Center back Bojana Popovic also scored six times for Montenegro.
Norway battled its way past Brazil, coming back from a six-goal deficit in the second half to win 21-19.
Two goals in the last four minutes by Camilla Herrem and Linn-Kristin Riegelhuth Koren helped Norway stay on course for its Olympic title defence.
"We were not on the gas, we were on the brake," Norway coach Thorir Hergeirsson said about the first 40 minutes of the match, in which his team was hit repeatedly by Brazil's quick and efficient counterattacks.
"Even at six goals down, we knew that if we could find our own concept, we could beat them," Hergeirsson said.
Top scorer Alexandra Nascimento scored five times to help Brazil build a 15-9 lead early in the second half, before Norway finally found its stride and denied the South Americans a first Olympic semifinal.
"We are realistic because we were playing the Olympic champions," said Nascimento. "We are sad as we were very close to a happy ending."
Norway has been the dominant force in women's handball in recent years, holding the Olympic, world and European titles simultaneously. But the team only qualified as fourth in Group B after losing to both France and Spain.
Hergeirsson said the weight of expectation was getting to his players.
"This must be the first step to do better," he said.
South Korea edged Russia 24-23 with Gwon Han-na scoring six times. South Korea won the title in 1988 and finished third at the Beijing Games.
"No, it's too early to think about the title, we still have two difficult matches," said right back Ryu Eun-hee. "We will relax a little now and focus just on the semifinal."
Left winger Elisabeth Pinedo Saenz scored seven times to help Spain defeat Croatia 25-22.
It's the first time that Spain, which won bronze at last year's world championships, reached the final four at the Olympics.
"I still can't believe we are here in the tournament," Pinedo Saenz said. "We took it step by step, it's incredible," Spain coach Jorge Duenas De Galarza said.
Neither team dominated the game until Croatia's key player Andrea Penezic injured her left ankle with her team leading 15-14 in the second half. When the right back briefly returned on the field, she found her team trailing 22-19.
"The Penezic injury was certainly a decisive factor," Croatia coach Vladimir Canjuga said. "We simply didn't have an equal substitute for her and our game was in disarray for a while."
The defeat ended Croatia's remarkable run in its first Olympics. It lost its opening Group A match against Brazil but recorded four straight wins afterward.