Dokic raced on to victory in 64 minutes to book a date with third seed Kim Clijsters.
Dokic faces Russia's Anastasia Myskina in the next round, who beat Italy's Francesca Schiavone 1-6 6-3 6-4.
The 19-year-old Yugoslav has played most tournaments unaccompanied by father and coach Damir Dokic over the past 18 months.
Alistair McCaw, a performance specialist to tennis stars including Jelena Dokic, has been watching the action close-up in Madrid.
Stevenson, who came through a gruelling three-setter against fifth seed Jelena Dokic on Thursday, looked tired and was often caught flat-footed.
Another feature match on Monday sees Australia's Jelena Dokic take on former world number one Ana Ivanovic on Pat Rafter Arena.
Dokic was also pleased with her performance after losing to Lindsay Davenport in the Bank of the West Classic quarter-finals last week.
The other upset of the day came in the women's draw, as third seed Jelena Dokic went down 7-6 3-6 6-3 to Amanda Coetzer.
Dokic, who had beaten second seed Jennifer Capriati on the way to the final, kept pace with the world number two for four games.
Dokic, chasing her third title of the year, raced out to a 3-0 lead against seventh seed Tanasugarn before eventually taking the opening set 6-3.
"I lost to (Tanasugarn) in the Hopman Cup two years ago but today I felt comfortable and I was dictating most of the points, " said Dokic.
In the first-round match between Martina Hingis and Jelena Dokic, the number one ladies' seed was knocked out by the 16-year-old and the score was an extraordinary 6-2, 6-0.
Two weeks after that she lost in the first round at Wimbledon to the unknown Jelena Dokic, with her mother again absent after Hingis had told her to stay away.
Kleybanova, who beat former world No. 4 Jelena Dokic in the first round, led 3-1 in the second set before her 27-year-old opponent rallied to win in two hours and 23 minutes.
She was playing with Meghann Shaughnessy against Dokic and Petrova but retired at the start of the final set and that has left a doubt hanging over her participation in the singles.
Ninth seed Marion Bartoli beat fellow Frenchwoman Virginie Razzano 7-5 6-0 and will next face Australia's former world No. 4 Jelena Dokic, who won 6-2 6-1 against 196th-ranked Russian Anna Chakvetadze -- a quarterfinalist in 2007 on her way to a career-high No. 5.
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