Just before the former Soviet ambassador, Anatoliy Dobrynin, left Washington to return to Moscow in April he reportedly called President Reagan and casually remarked: "You know, you really ought to deal more with us through Ambassador Arthur A. Hartman."

This anecdote, told by an administration official to a third party, in retrospect appears to have been a harbinger of Dobrynin's ambitious plans to keep a tight grip on his government's main American connection by channeling communications through Hartman and himself in Moscow.

In what many U.S. specialists here regard as a major change in the way the Soviet Union does business, Dobrynin has now become head of the Central Committee's international department with an apparent mandate from Soviet leader Mikhail Gorbachev to remain a key player in the shaping of Soviet policy toward the United States.

These specialists find confirmation of this thesis not only in Dobrynin's unusual new posting but also in the appointment of Yuri Dubinin, a career diplomat who speaks little English and has had no experience in U.S. affairs, to replace him as Moscow's ambassador to Washington.

"Dobrynin didn't want a Dobrynin in Washington while he is making American policy in Moscow," remarked Arnold Horelick, director of the Rand Corp.'s Soviet studies program and formerly the Central Intelligence Agency's national intelligence officer for the Soviet Union.

"Dobrynin is trying to channel things through Moscow and avoid a back channel developing through Washington," said Bruce Parrott, director of Soviet studies at Johns Hopkins School of Advanced International Studies.

Dubinin, who presented his credentials Monday to President Reagan and quickly made his first calls on Congress and the State Department, has already impressed administration officials as a diplomatic "pro," a man who on his first outing in Washington made "no trips, falls or bumps on his head," as one put it.

But no one seems to think Dubinin will have the weight in Moscow for years to come to match that of Dobrynin, who spent 24 years in Washington and is acknowledged as Moscow's top specialist in American affairs. With his recent promotion, Dobrynin is now a member of the senior leadership of the Soviet Union, standing just below members of the ruling Politburo.

"For a long time, the show is going to be run by the people who know this country the best," remarked the administration official. "Dubinin is very much the new boy."

Some U.S. Soviet specialists view Dobrynin's new role in Moscow as part of Gorbachev's grand design to try to deal with the United States through political means rather than through a new arms race and a continuing military buildup.

"Gorbachev would prefer to put much more emphasis on diplomacy than an all-out arms race for domestic reasons," said Parrott, referring to the Soviet leader's preoccupation with the Soviet economy.

Another example of the new Gorbachev diplomacy surfaced here last week when Dubinin delivered to the Senate a proposal for a meeting "in the nearest future" between the two foreign affairs committees of the Supreme Soviet, the Soviet Union's legislative body, and those of the U.S. Congress. Since Dobrynin is head of one of the Soviet committees, he would be dealing once again directly with Congress, whose leaders he knows well.

"The idea of the meeting had Dobrynin's fingerprints on it," a Senate Democratic aide said.

U.S. Soviet specialists said the appointment of Dobrynin to a senior post in the Central Committee's bureaucracy as head of its international department could have far-reaching implications. It holds the potential, they said, for a major change in the way Soviet foreign policy has been made over the past three decades.

From his elevation to the Politburo on April 27, 1973, until June 1985, former foreign minister Andrei Gromyko was Moscow's "czar" when it came to dealing with the United States. With his replacement by the inexperienced Eduard Shevardnadze, the Foreign Ministry is no longer as central as it once was in handling American affairs, these specialists said.

Furthermore, by making Dobrynin the head of the Central Committee's international department, Gorbachev has propelled the party apparatus into the forefront of foreign policy and seemingly created a built-in bureaucratic struggle between it and the Foreign Ministry, they said.

"There will be a certain rivalry and pulling and hauling between the Foreign Ministry and the Central Committee now," Raymond L. Garthoff, a Soviet specialist at the Brookings Institution, said.

Further complicating relations between Shevardnadze and Dobrynin, according to U.S. specialists, is the fact the new foreign minister formally outranks the former envoy to Washington because Shevardnadze is a full member of the Politburo.

Just what Gorbachev had in mind by making Dobrynin secretary of the Central Committee and head of the party's international department is the topic of much speculation among specialists. But it is already clear the international department will no longer be dealing just with Soviet relations with other nations' communist parties and with Third World nations as it did under Dobrynin's predecessor, Boris Ponomarev.

Dobrynin has surrounded himself with a number of top Soviet foreign affairs specialists, including Georgi Kornienko, who was formerly first deputy foreign minister and a noted specialist in U.S. affairs and arms control, and Yuli Vorontsov, former ambassador to Paris and an earlier minister in Washington under Dobrynin. Another of Dobrynin's deputies is Karen Brutens, a longtime specialist in Third World affairs.

Some U.S. Soviet specialists suggested that Dobrynin may have in mind as a model the White House's National Security Council (NSC), which he knew well, particularly when its staff was run by Henry A. Kissinger in the Nixon administration. Garthoff warned, however, that while there may be some superficial similarities between the NSC and the new international department in Moscow, "the comparison may mislead us more than help us in understanding its role."

The Soviet party's international department, Garthoff and several other specialists noted, has no coordinating role -- at least not yet -- over the various Soviet ministries and agencies, which is one of the NSC's chief functions. In particular, Dobrynin's department has no control over the Soviet military establishment.

Another unknown is what role Dobrynin, with his new hat, will play in drafting Soviet arms control proposals. Garthoff said he believes Dobrynin is already "actively involved" in that process but notes that a number of the new Soviet proposals were put forward before Dobrynin returned to Moscow.

"It's very clear they Dobrynin and his staff will be doing a lot more in this area. But how it will work, we'll have to see," Garthoff said.