The Australian Open champion, who now faces Poland's Agnieszka Radwanska, grabbed a hard-fought first set but had to fight off a sustained challenge from Barthel, who had eliminated former world number one Ana Ivanovic and world number seven Mario Bartoli of France in the previous rounds.
"She showed some great tennis today," Azarenka said. "I always said since the start of the year she is the one to watch."
The Belarussian has lost only once this season after going 26 matches without defeat.
"I am trying to become better and better. This is new for me (being number one) but I feel good and confident being there."
The pair traded punishing blows throughout the second set with Barthel twice moving to within two points of winning the set, at 5-4 and 6-5, before Azarenka briefly pulled herself out of the hole.
Fired on by the local crowd who mere mesmerised by her mix of raw power and intelligent play, big-serving world number 35 Barthel, who won her maiden WTA tour event in Hobart in January, grabbed the second set on her first set point.
Another huge backhand winner down the line gave her the first break opportunity at the start of the second set and Azarenka cracked under pressure to send the next point long.
But she recovered to twice break back and, after squandering a match point at 5-4, finished her opponent off two games later.
Azarenka's next opponent, fourth seed
Na looked mightily impressive in the opening set as she found a fine rhythm on her groundstrokes to take it in convincing fashion.
But the eighth seed was not able to sustain her momentum as Radwanska roared back to take the second set for the loss of just two games.
Na suddenly lost her composure and her groundstrokes appeared apprehensive and devoid of conviction.
Radwanska was not to be denied in the final set, and the Pole took it 6-3 after a flurry of belligerent shots from the back of the court.
Later on Friday, world number two
Stosur edged the opening tie-break after a fiercely-contested first set, and Sharapova looked to be under real pressure at the start of the second.
But the Russian held her composure to take the set to another tie-break, and this time she prevailed to take it and level up the match.
The third set went Sharapova's way courtesy of a solitary break of serve, and the world number two will next take on Wimbledon champion