Wanwrika served for the match at 5-4 in the second set, but Becker broke with two straight forehand winners. The 137th-ranked German saved two match points when serving at 5-6, and soon won with a backhand winner.
The loss damaged Wawrinka's chances of qualifying for next month's Masters Cup in Shanghai. The 23-year-old Swiss is 11th in the standings, and only the top eight qualify.
Also competing at Basel are Juan Martin del Potro of Argentina, who is eighth, and 10th-place James Blake of the United States.
Eighth-seeded Mardy Fish of the United States also advanced, beating Agustin Calleri of Argentina, 7-6 (5), 6-2, while Feliciano Lopez of Spain defeated Swiss wild card Marco Chiudinelli 7-6 (5), 7-6 (7) and sixth-seeded Igor Andreev of Russia beat Juergen Melzer of Austria 7-6 (5), 7-5.
Two-time defending champion Roger Federer, playing in his hometown, opens Tuesday against Bobby Reynolds of the United States.
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