Sharapova had control in her next service game, but Li scrambled from side to side and pushed the reigning French Open champion to go for the lines, getting a series of unforced errors and another break.
The sixth-seeded Li has been working since August with Rodriguez, and credits him with reviving her career with a renewed emphasis on condition.
‘‘I’m happy. I know I have a tough coach, a tough physio,’’ Li said, looking across to the stands and adding: ‘‘You don’t need to push me anymore. I will push myself.’’
Sharapova, who lost the 2012 Australian final in straight sets to Azarenka, admitted it was hard to get into the match against Li.
‘‘She was certainly much more aggressive than I was, dictating the play. I was always on the defense,’’ said Sharapova, who could have gained the No. 1 ranking by reaching the Australian final.
Djokovic was stretched to five sets in a 5-hour, 2-minute fourth-round match against Stanislaw Wawrinka, then lost a set against Tomas Berdych in the quarterfinals on Tuesday.
But the hard-serving Djokovic was at his sublime best on Thursday night, converting all of seven break-point chances while not facing any on his serve. Now unbeaten in five Grand Slam head-to-heads with Ferrer, he allowed his opponent only four points in the first four games of the last set.
‘‘I felt very comfortable and very confident from the start,’’ he said. ‘‘I wanted to be aggressive on the court. This is definitely one of the best performances I ever had in my career.’’
Ferrer double-faulted to set up match point, then lost the match on the following point when he floated a backhand long.
The Spaniard was pleased to have made it as far as he did in the tournament.
‘‘I prefer to play worse in a semifinal than in the first round,’’ Ferrer said. ‘‘Of course, I am not happy with my game tonight, but this is tennis.’’
With such a light workout during his match, Djokovic went back out to center court at Rod Laver Arena an hour later during a legends doubles match involving his friend, retired Frenchman Henri Leconte.
To much laughter from the crowd, Djokovic, dressed in a mock medical costume, pretended to give Leconte treatment at the side of the court.
Later, he was all business while talking about the final against either Federer or Murray.
‘‘It can only do positive things to my confidence at this stage of the tournament,’’ Djokovic said of his easy win. ‘‘I have a great feeling about myself at the moment.’’