The Press, Cleveland's afternoon newspaper, moved into direct competition with the city's dominant morning paper this week by publishing its first Sunday edition in an effort to stop a steady decline in circulation.

While executives at the morning Plain Dealer may be skeptical, the Press says the results of this marketing move should be heartening for employes of other afternoon dailies discouraged by the announcement that Washington's afternoon paper would cease publishing Aug. 7.

"The town is just turned on. Our phones are ringing with advertisers," said Joseph E. Cole, the millionaire businessman who bought the Press from the Scripps-Howard organization last November, after the paper had been losing money for several years.

About 30,000 copies of the Press' 96-page Sunday edition were sold at 25 cents at newsstands and in stores. Eighty-five percent of the stores were sold out by mid-morning, said Gerald Gordon, general manager of the Press.

But William C. Barnard, managing editor of the Plain Dealer, while acknowledging his paper was "certainly concerned" about increased efforts by the Press, noted that most of the 425,000 Sunday editions of the Press were circulated as samples or simply given to those who already had Monday-through-Saturday subscriptions.

The Press' Sunday edition relied heavily on bold graphics and full-color photographs.

"Naturally we don't smirk at any effort that the Press makes," Barnard said, "but what they have to offer is substantially different from what the Plain Dealer has." He said many of the Sunday Press' articles were "soft" features.

The Press, which has been losing money since the early 1970s, lost $6 million last year. Circulation slipped from 320,000 daily in 1978 to 302,410 this spring. The Plain Dealer's circulation increased during that time from 379,400 to 406,400.

Although Cole declined to state the added cost of producing the Sunday edition, he said he has invested more than $3 million in new equipment since taking over the Press.

Herbert Kamm, editor of The Press, said there is no comparison between the situations of the afternoon papers in Washington and in Cleveland. "They're entirely different communities," he said.

Home delivery is less of a problem in Cleveland, he said, saying that "Washington's a hard town to get around in."