Smith and Skupski looked to be heading for a valiant second-round exit to Croatia’s Ivan Dodig and Sania Mirza of Indonesia when they trailed 5-2 in the third and final set.
However, the Brits then reeled off five successive games – saving four match points along the way – to claim an outstanding 4-6 6-3 7-5 victory in front of a delighted crowd on Court 18.
Barry Scollo, Director of Tennis at Team Bath, said: “That may be our best-ever result in doubles, what a performance by Anna and Neal.
“Mirza has been one of the best doubles player in the world during the past two years, certainly in her ladies’ doubles partnership with Martina Hingis.
Smith had overcome a familiar face in the first round of the mixed competition on Monday, beating ladies’ doubles partner Jocelyn Rae and Colin Fleming 6-1 1-6 6-3.
The third round will see Smith and Skupski face either Julian Knowle and Shuai Zhang or number 14 seeds Aisam Qureshi and Yaroslava Shvedova.