The 45-year-old Swiss-Italian general secretary of UEFA got 115 votes in the second round of the election held at the FIFA Congress while the Asian Football Confederation president got 88.
Infantino takes over the 209-member, multi-billion dollar body from Sepp Blatter with world football's image at an all-time low.
Blatter has been banned from football for six years and US and Swiss authorities have launched major corruption investigations into FIFA and other football federations.
"And we will restore the image of FIFA and the respect of FIFA and everyone in the world will applaud us," a visibly surprised Infantino told the Congress.
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Sheikh Salman, a member of the Bahrain royal family, was one of the first to congratulate his rival.
Having got 27 votes in the first round, Jordan's Prince Ali bin al Hussein got just four in the second.
Former FIFA official Jerome Champagne had seven votes in the first round but zero in the deciding ballot.
The Congress was given tough warnings before the vote that the result and a series of agreed reforms had to convince a world shocked by several years of corruption scandals and doubts about World Cup bidding.
International Olympic Committee president Thomas Bach said football and all sports had to improve governance.
"Today you have this great chance to turn the page," he told the congress. "We need to give new answers to the new questions with regards to credibility and good governance."
Earlier a reform package was adopted by 179 members, while 22 voted against and six abstained.
The president's job has been altered to function like a corporate chairman of the board, providing strategic guidance but with less management authority.
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Measures such as declaring the salary of the new president to improve financial transparency were also included.
After several years of suspicions, scandal erupted at FIFA when seven top officials were arrested at a FIFA congress in May last year.
Infantino will face immediate financial problems.
Acting Secretary general Markus Kattner said "general uncertainty" following the crisis had created tough economic times and FIFA was roughly USD 550 million (500 million euros) behind in its USD 5 billion budget plan for 2015-2018.
Sheikh Salman, who is seen as closer to the FIFA old guard, had said the proposal could bankrupt FIFA.
Infantino countered that his proposals were far from reckless and that his record at UEFA proved his credibility as a financial manager.
"When I speak about figures...I know what I am speaking about," he said.
Blatter, 79, was the big absentee at the congress. The Swiss sports baron suffered a spectacular fall over the last nine months.
Blatter has since been banned from football for six years for ethics breaches and could face criminal charges.
Each of the rivals went into the vote with political problems. Infantino was for seven years the right-hand man of Michel Platini, the UEFA president also banned for six years for ethics breaches.
Sheikh Salman had faced tough questions about the clampdown on pro-democracy protests in the Gulf state. He has called allegations made by human rights groups "nasty lies.