Tilghman announced in the fall her intention to step down at the end of this academic year after 12 years as president at the Ivy League school of more than 8,000 undergraduate and graduate students.
Eisgruber, 51, said he was honored to lead the university, from which he graduated in 1983 with a degree in physics.
"This university has shaped my life ever since I set foot on this campus as a freshman 34 years ago," he said, "and that time as a student, as an alumnus, and as a faculty member has given me a heartfelt appreciation for Princeton's very special strengths."
Gov. Christie attended the announcement and later said he was happy to see Eisgruber advance to the role of president.
"What strikes me is how much he loves this place," Christie said, adding that when Hall asked for his advice on the search, he said, "Get someone who loves this place . . . because of the complexities of this university."
As governor, Christie is an ex-officio board member. His oldest son, Andrew, is a freshman at the school.
Eisgruber spoke of the challenges ahead for the university, including the cost of attending, online education, and defining the importance of a liberal arts education.
He said that over the last four decades, Princeton had kept a commitment to greater inclusiveness and access.
"That commitment has many manifestations, including our unsurpassed financial-aid program," Eisgruber said. "But we also know this is an area where we have more work to do."
An education at Princeton is "a gift, one that can transform the life of any student, faculty member, or other scholar who is lucky enough to receive it," Eisgruber said, adding that the university had an obligation to make that gift accessible.
In calling Princeton "the world's best liberal arts college," Eisgruber - alluding to the debate over the relevance of a liberal arts education - said he believed it was more important now than ever.
That is also, in part, why he wants to continue to explore online education, addressing, he said, two specific areas:
"What does the advent of online education mean to Princeton, and how do we wish to participate in it?"
Said Eisgruber: "Princeton thrives on challenges, and I have no doubt we will thrive on these challenges."
After graduating from Princeton, Eisgruber, an Oregon native, spent two years at Oxford University as a Rhodes Scholar, and then earned a law degree from the University of Chicago.
He clerked for U.S. Supreme Court Justice John Paul Stevens and taught at the New York University School of Law for 11 years before joining the Princeton faculty in 2001 as the director of the Program in Law and Public Affairs.
Eisgruber was named Princeton's 11th provost in 2004, serving as the university's chief academic and budgetary officer. He lives in Princeton with his wife, Lori A. Martin, and their 14-year-old son, Danny.
Hall, the trustees chairwoman, said Eisgruber's leadership during the depths of the recession in 2008 and 2009 was "absolutely critical" and also lauded his transparency with the university's finances. The departing president agreed.
"In Chris we have the leader that we are going to need for the next decade or so, and I don't think we could be in better hands," Tilghman said.
Contact Claudia Vargas at 856-779-3917, email@example.com, or follow on Twitter @InqCVargas. Read her blog, "Camden Flow," at www.philly.com/camden_flow.