Mr. de Blasio spoke of being "honored" to work in the Clinton administration as a regional director of the Department of Housing and Urban Development and "inspired" to serve as the campaign manager when Mrs. Clinton ran for the Senate in 2000.

When Mr. de Blasio asked Mr. Clinton to swear him in as the city's 109th mayor, it was a nod to the pragmatic approach of the former president. And in turn, the Clintons' prominent presence brought the former first couple a progressive sheen as the Democratic Party shifts to the left, and Mrs. Clinton contemplates another run for president in 2016.

Throughout his campaign and at his inauguration, Mr. de Blasio did not sound like a Clinton Democrat, but as his political star rose, he consistently reinforced his connection to the Clintons.

Harold M. Ickes, who was a deputy chief of staff in the Clinton White House, advised Mr. de Blasio on his campaign and served on his transition team. Mr. de Blasio posted a photograph from 2000 on Twitter of his family with Mrs. Clinton and a younger, plumper Mr. Clinton.

After he sealed the Democratic nomination, Mr. de Blasio told reporters he received phone calls from both Mr. and Mrs. Clinton, who also endorsed him. "They both offered extraordinarily helpful advice," Mr. de Blasio said, adding: "I am proud to come from the Clinton family."

But until that endorsement and a related fund-raiser, the de Blasio-Clinton love fest seemed largely one-sided.

Although Mr. de Blasio has known the Clintons for years, they do not have a close relationship and Mr. de Blasio is not considered to be in their inner circle, according to several people who know both families and who spoke on the condition of anonymity so they could talk candidly about the relationship. Even when Mr. de Blasio ran Mrs. Clinton's successful Senate campaign in New York, he found himself, as many Clinton newcomers have, answering to the first lady's cadre of loyal advisers.

But the Clintons take pride in Mr. de Blasio's success and count him among their extended family of political disciples. That circle is expansive and includes the mayor of Chicago, Rahm Emanuel, and many members of President Obama's administration. "It makes sense the Clintons would feel like a part of this," said Representative Charles B. Rangel, a Democrat from Harlem.

But that does not mean Mr. de Blasio has forged his policies in the same centrist vein as Mr. Clinton. "There's nothing to compare him to, quite frankly," Mr. Rangel said.

In his brief speech introducing Mr. de Blasio, Mr. Clinton strongly endorsed the mayor's "core campaign commitment that we have to have a city of shared opportunities, shared prosperity, shared responsibility," and called the growing tide of economic inequality a "horrible constraint on economic growth."

Mr. de Blasio then thanked the Clintons, who sat in a row next to Gov. Andrew M. Cuomo and across the aisle from the de Blasio family.

"Over 20 years ago, when a conservative philosophy seemed dominant, you broke through and told us to still believe in a place called Hope," Mr. de Blasio told Mr. Clinton. He then praised Mrs. Clinton's "groundbreaking commitment" to early childhood education, a priority in Mr. de Blasio's own campaign.

After an hour of poems and prayers as well as speeches that repudiated the policies of former Mayor Michael R. Bloomberg, Mr. Clinton was the first speaker to strike a more conciliatory tone.

"I also want to thank Mayor Bloomberg, who has committed so much of his life to this city," Mr. Clinton said. "He leaves the city stronger and healthier than he found it."

The Clintons had remained neutral in the Democratic primary partly because Anthony D. Weiner, the husband of one of Mrs. Clinton's closest aides, Huma Abedin, was also a candidate. Many in Mrs. Clinton's camp also had ties to and affection for Christine C. Quinn, the former City Council speaker. (Ms. Abedin attended the inauguration, escorting Mrs. Clinton to her chair on stage.)