Teenager Arthur Fery and fellow Briton Anton Matusevic reached the final round of men's qualifying at Wimbledon.
Bains, 23, came from 4-1 down against Hungary's Reca Luca Jani to win 7-5 6-3.
Similarly, 22-year-old Swan trailed Australia's Lizette Cabrera 4-2 before overhauling the deficit and winning 7-5 6-2.
Matusevich fended off break points in his opening game before breaking at the end of a competitive first set for the lead over Spain's Carlos Taberner.
The 20-year-old world number 454 dominated the second set on his way to a 6-4 6-1 win, and he will face 10th seed Bernabe Zapata Miralles, also from Spain, in the final round.