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Wawrinka recorded his fifth victory over Kohlschreiber in as many meetings in a performance that left him cautiously encouraged about his prospects of making it past the quarter-finals in the California desert for the first time.
"It was a really good match," said Wawrinka, who owns three Grand Slam titles but has won just one of the tour's coveted Masters titles.
"The first set was not easy, for sure, but in general I'm happy with my game. I was serving well. I think I'm moving better and better and it's all positive so far."
A break in each set was enough for Wawrinka, who didn't face a break point himself as he advanced to a meeting with lucky loser Yoshihito Nishioka of Japan, who staged a remarkable rally to get past 13th-seeded Czech Tomas Berdych 1-6, 7-6 (7/5), 6-4.
The 21-year-old from Mie saved a match point against his own serve in the 10th game of the second set.
After Nishioka forced the decisive set by winning the tiebreaker, the two traded breaks to open the third before Nishioka broke for a 2-1.
Although Berdych saved one match point in the ninth game, Nishioka closed it out in the next with a love game to reach the fourth round of a Masters tournament for the first time.
"It's going to be interesting. I've never practiced with him, so it's going to be the first time we going to be against each other."
Wawrinka is the highest-ranked player remaining in the top half of the draw after world number one Andy Murray's shock loss to 129th-ranked qualifier Vasek Pospisil in the second round.
With superstars Novak Djokovic, Rafael Nadal and Roger Federer all jammed into the bottom quarter of the draw, Wawrinka's path looks wide open.
But the Swiss said he didn't think of it that way.
"It's not like I have been in a lot of semi-finals, finals in Masters 1000," he said. "So I need to focus on every match."
Eighth-seeded Austrian Dominic Thiem powered into the fourth round with a 6-1, 6-4 victory over Germany's Mischa Zverev.
World number nine Thiem, riding high after winning the ATP title in Rio in February, had little trouble from Zverev's serve-and-volley style.
Although the German saved five match points in the ninth game of the second set, Thiem served out the victory in the next game in less than an hour.
Thiem next faces 10th-seeded Gael Monfils, who dispatched American John Isner 6-2, 6-4 in just 68 minutes.
Elsewhere, Pospisil was unable to build on the best win of his career falling to fellow qualifier Dusan Lajovic 7-6 (7/4), 3-6, 7-5 just two days after toppling Murray.
Pospisil served for the match at 5-3 in the third set but was broken to love, then failed to convert a match point in the 10th game as Lajovic won the last four games to take the match.
Results on Monday in the BNP Paribas Open ATP Masters/WTA tournament in Indian Wells, California (x denotes seeded player):
Men
Third round
Dusan Lajovic (SRB) bt Vasek Pospisil (CAN) 6-7 (4/7), 6-3, 7-5
Pablo Carreno-Busta (ESP x21) bt Roberto Bautista (ESP x16) walkover
David Goffin (BEL x11) bt Albert Ramos (ESP x22) 7-6 (7/3), 6-4
Pablo Cuevas (URU x27) bt Fabio Fognini (ITA) 6-1, 6-4
Stanislas Wawrinka (SUI x3) bt Philipp Kohlschreiber (GER x28) 7-5, 6-3
Yoshihito Nishioka (JPN) bt Tomas Berdych (CZE x13) 1-6, 7-6 (7/5), 6-4
Gael Monfils (FRA x10) bt John Isner (USA x20) 6-2, 6-4
Dominic Thiem (AUT x8) bt Mischa Zverev (GER x29) 6-1, 6-4
Women
Third round
Madison Keys (USA x9) bt Naomi Osaka (JPN) 6-1, 6-4
Caroline Wozniacki (DEN x13) bt Katerina Siniakova (CZE) 6-3, 6-1
Lauren Davis (USA) bt Julia Goerges (GER) 6-1, 6-4
Kristina Mladenovic (FRA x28) bt Simona Halep (ROM x4) 6-3, 6-3
Peng Shuai (CHN) bt Agnieszka Radwanska (POL x6) 6-4, 6-4
Venus Williams (USA x12) bt Lucie Safarova (CZE) 6-4, 6-2
Elena Vesnina (RUS x14) bt Tímea Babos (HUN x25) 6-4, 1-6, 6-4
Angelique Kerber (GER x2) bt Pauline Parmentier (FRA) 7-5, 3-6, 7-5
