Beaumont prevailed by a 28-23 margin to secure a second consecutive four-year term.
Frenchman Bernard Laporte, who was running unopposed, has been elected vice-chairman.
"Now is not the time for celebration. We have work to do," Beaumont said in a statement.
"We are tackling COVID-19 and must implement an appropriate return-to-rugby strategy that prioritises player welfare, while optimising any opportunity to return to international rugby this year in full collaboration with club competitions for the good of players, fans and the overall financial health of the sport."
Rugby, like most sports, has been on hold as the novel coronavirus spread across the world, casting doubt on whether the 2020 international calendar could be completed.
"I am determined to ensure that the spirit of unity and solidarity that has characterised our work in response to an unprecedented global COVID-19 pandemic is the cornerstone of a new approach that will deliver a stronger, more sustainable game when we emerge with new enthusiasm, a renewed purpose and an exciting future," Beaumont added.
Pichot ran against Beaumont, after serving as his vice-chairman, and had the support of New Zealand Rugby and its SANZAAR partners.
"I would like to thank Gus for his friendship and support over the last four years. While we stood against each other in this campaign, we aligned in many ways, and I have the utmost respect for him," Beaumont said.
"Gus is passionate about the sport and his contribution has been significant."
Newly appointed New Zealand Rugby board member Bart Campbell has been elected to World Rugby's new executive committee.