It was a "happy coincidence" said Archbishop Pio Laghi, that he was able to begin his tenure as the Vatican's official representative to the United States the same week that a new president was sworn in and the country erupted in a national celebration of the hostages' return.

For the new apostolic delegate, his arrival here Jan. 18 was a return to the country in which he began his career in the Vatican diplomatic service 20 years ago. But the last two decades, he says, have been a period of intense change in this country and even more so in the church.

Chatting with visitors at the stately delegation headquarters at 34th Street and Massachusetts Avenue, Laghi spoke both of changes fostered by the Second Vatican Council and by the passing of church leadership from Pope Paul VI to Pope John Paul I to John Paul II. He cited particularly the changes signaled by the present pontiff.

"In those three months [between the death of Paul and the accession of John Paul II] the church in various parts of the world has progressed, in some places, 30 years," he said.

Laghi, who comes here from a similar post in Argentina, made it clear that his mission here was primarily a religious one. "My task will be for the church and in the church," he said. "I will be an alive sign of connection and solidarity between the Catholic hierarchy and the pope."

Laghi, 58, is a trim, silver-haired Italian who speaks English fluently, although with an occasional appeal to his listeners for help in finding the precise word. He succeeds the Belgian-born Archbishop Jean Jadot, who left in September to assume a Vatican post after seven years here.

Jadot is generally credited with changing the face of the American Catholic Church through his part in the selection of 174 bishops. The "Jadot bishops," as they have come to be known, tend to be young and progressive with a commitment to implementation of the Vatican Council reforms. His choices were not always popular with conservatives in the church.

Laghi, who served in Jerusalem, India and Nicaragua before his assignment to Argentina, said he had "a long conversation with Jadot" before coming here, but he indicated he intended to put his own stamp on the office. "My mission here is not to be a continuation of the mission of Archbishop Jadot. It is the beginning and continuation of Archbishop Laghi's mission."

In addition to his post as apostolic delegate, Laghi will function as Vatican Observer to the Organization of American States. He told questioners that he would express the concern of the American hierarchy about the situation in El Salvador in that forum. "I really am very concerned about the situation there," he said.

Laghi said he expects to follow the example of Pope John Paul II and spend a fair amount of time traveling around this country. This week he was in Dallas, where about 200 bishops gathered for a conference on sexuality.

Laghi's introduction to the rest of the Washington embassy community and to the new American leaders began at a White House reception President and Mrs. Reagan hosted for the diplomatic corps.

"There I was, the last in the line [because he was the diplomat most recently accredited here]; there they were in their national dress, while I" -- he smiled as his fingered his crisp, white clerical collar -- "was in my international costume."