Broady, 25, defeated Croatian Ajla Tomljanovic 6-3 6-4, while Konta was even more emphatic in dispatching Australia’s Jarmila Gajdosova 6-1 6-3 in less than an hour.
Broady will take on Romania’s world number three Simona Halep in the next round and said: “I’m excited about it. I’ve got no pressure on me whatsoever. If I serve well, who knows what’s going to happen.
“I don’t think grass is her favourite surface of all of them, so we’ll wait and see. I’m just going to aim to go out there and play my game and take it as close as I can and go for it really.”
Konta, who beat the world number 60 in Gajdosova, said: “She’s a great player and she has a great weight of shot, so I went out there to do my best and I’m very happy that I came through the match.”
A third Briton, 16-year-old Katie Swan, was beaten 6-2 6-2 by Bulgaria’s Tsvetana Pironkova.
Ninth-seeded Spaniard Garbine Muguruza and Alize Cornet, the 11th seed from France, were surprise opening-round casualties.
Slovakia’s Magdalena Rybarikova was a 6-3 6-1 winner over Muguruza, while Cornet went down 4-6 6-2 6-2 to Christina McHale of the United States.
Other first-round winners included Czech trio Lucie Hradecka, Katerina Siniakova and Klara Koukalova, Australia’s Casey Dellacqua, Slovakia’s Anna Karolina Schmiedlova and Portugal’s Michelle Larcher de Brito.