The Japanese fourth seed, who had to fight back from a set down against Hsieh Su-Wei in her previous match, was far from her best.
She struggled to hold serve and berated herself regularly during the sun-bathed clash at Rod Laver Arena.
She held firm when it counted, however, breaking Sevastova in the seventh game of the decider.
She then sealed the match after one hour and 47 minutes with an imperious forehand down the line.
Osaka will play sixth seed Elina Svitolina for a place in the semi-finals.