The 23-time grand slam champion leads Wawrinka 21-6, head-to-head. This was their just third meeting since the final of the 2016 US Open, which the Swiss player won.
Tiafoe, who defeated Wu Yibing on Wednesday to start his campaign, maintained his excellent play to reach the third round for the fifth consecutive major tournament. The American won the final two sets after fending off four set points in a lengthy opening tie-break.
Murray is chasing his second ATP Challenger title for this year, having captured the Challengers 175 crown at Aix-en-Provence, France by defeating Tommy Paul in the final