Very big. While the "space race" term now usually is applied to the moon landing, the "first man in space" achievement was also considered a marker.

When he flew, my first impression was - well, they beat us again.

-- Charles Duke, astronaut from the Apollo 16 mission

The New York Times had a giant, splashy, front page: SOVIET ORBITS MAN AND RECOVERS HIM; SPACE PIONEER REPORTS: 'I FEEL WELL'; SENT MESAGES WHILE CIRCLING EARTH

and one of the articles starts with

The Soviet Union announced today it had won the race to put a man into space.

Here are some other front pages courtesy the Columbia Journalism Review:

Man Leaps Free of Earth Shackles (Christian Science Monitor)

Soviet Man Orbits, Returns to Earth (Milwaukee Journal)

Soviet Spaceman Lands: 'I FEEL WELL' (New York Post)

The import of the announcement was not all politics and competition: it really was considered a wonder. The political cartoonist for the Washington Post (Herbert Block) depicts the majesty of the achievement.

There is an ABC News Special from April 12 (video link here), calling the moment one of the most memorable of the century, with quotes like (in reference to Gagarin) "he has now traveled faster, higher, and farther than any man in the history of this planet" and agreement with the assessment that it was "an unparalleled victory of man over the forces of nature".