When Mr. Rather was told that the tennis match between Steffi Graf and Lori McNeil was closer than expected and that the ''Evening News'' would be truncated, he telephoned New York to protest to Howard Stringer, the president of CBS News, according to several CBS staff members.

Mr. Rather told Mr. Stringer that if his broadcast did not go on at 6:30 as scheduled, he would not be in his anchor chair and that CBS Sports should fill the rest of the half-hour, according to the staff members and CBS officials. Mr. Stringer then telephoned Gene Jankowski, the president of the CBS Broadcast Group, to negotiate an earlier end to the tennis coverage.

Mr. Stringer apparently succeeded in getting some of the time back, but the sports crew in New York still needed a couple of minutes of ''Evening News'' time to wrap up their coverage.

It has been a difficult summer for Mr. Rather, whose broadcast slipped into third place in the ratings after nearly five years in first place. There has been heated discussion inside CBS News over the past few months about proposed changes in the Evening News, and there was for a time speculation that a co-anchor would join Mr. Rather on the broadcast. But last week, a new ratings system took effect, and it showed Mr. Rather back in first place, and many in the organization hoped that the broadcast was past its difficulties. Rather Left Studio

When 6:30 came, Mr. Rather was in his anchor chair, his microphone on and ready to go. But when CBS Sports did not yield the network air, Mr. Rather disconnected his microphone and walked out of the studio, according to CBS people in Miami.