"Given the unusual outcome in relation to the results-expectation and head-to-head history, and in the interests of maintaining unequivocal confidence in our game, FIFA Security will conduct a routine examination of this game and its result," FIFA said in a statement.
Bahrain had needed a nine-goal win in the Asian Group E match, combined with a defeat for Qatar in Iran, to progress to the next stage of the region's World Cup qualifying competition.
Qatar went through, however, when an 86th-minute goal gave them a 2-2 draw in Iran to clinch second place in the group at Bahrain's expense.
Indonesia, already eliminated, fielded a weakened team as a number of their top players had been suspended by their own federation for taking part in the breakaway Indonesian Super League (ISL).
Bahrain converted two penalties during the match and Lebanese referee Andre El Hadded sent off Indonesia goalkeeper Samsidar after two minutes.
According to FIFA's records, the two sides had met six times before with two wins apiece and two draws. Bahrain had not previously won by more than two goals.
Indonesian football has recently been in turmoil and the country came close to international suspension last year.
Djohar Arifin Husin, a former secretary-general of the Indonesian Sports Committee, was elected head of the FA (PSSI) last year, defeating two other candidates in elections ordered by FIFA.
'HUGE ISSUE'
Previous elections had failed amid chaotic scenes and FIFA appointed a committee last year to organise the election.
The previous PSSI president, Nurdin Halid, was jailed in 2007 for misusing funds but continued to run the FA from his cell.
Prince Ali of Jordan, FIFA's Asian vice-president with a seat on its executive board, told reporters in London before this weekend's International Board meeting that while he was aware of the surprising result, he could not comment directly on it.
However, he added: "Match fixing is a huge issue that needs to be tackled... we will have to see what the investigation comes out with.
"But it has to be taken very seriously, regardless of what region it is played in. It is a world issue, not just simply in the Asian region. Many times you see those involved are a step ahead," he said.
"We need to put as many resources as we can into this aspect of football and support those who are dealing with it in FIFA.
"The important thing is that if there are suspicions you have to investigate it. It might just be a coincidence, however there might be something behind it. Regardless, it can happen in any country in the world."
