Stosur and Stephens – seeded second and third respectively – were never troubled in their straight-sets wins in Birmingham.
Australian Stosur hit eight aces in a 6-1 6-3 win over American Christina McHale that took her just 65 minutes to complete on the grass on Wednesday.
The 2011 US Open champion won 34 of 45 points on serve and made four breaks to set up a meeting against compatriot Casey Dellacqua at the WTA International event.
Dellacqua also beat an American – Varvara Lepchenko – 6-4 6-3.
Third seed Stephens beat Italian Francesca Schiavone 6-2 6-4 in a scoreline that was perhaps unfair on the loser.
Schiavone won 46 per cent of points in the match, but, crucially, failed to take any of the five break points she set up.
Stephens is due to play fellow American Alison Riske in the third round.
Riske defeated Ukrainian qualifier Nadiia Kichenok 7-6 (7-4) 6-1.
The shock of the day saw Safarova knocked out by fellow Czech Barbora Zahlavova Strycova, who won 6-3 3-6 7-5.
Other seeds to lose were American Madison Keys, Monica Puig of Puerto Rico and Slovakian Magdalena Rybarikova.
Hungarian Timea Babos thrashed Keys 6-2 6-1 while Japanese veteran Kimiko Date-Krumm came from behind to beat Puig 2-6 6-4 7-6 (9-7).
And Czech Petra Cetkovska toppled Rybarikova 6-4 7-5.
Fourth seed Kirsten Flipkens of Belgium made four breaks in a 6-3 6-2 triumph over Italian Camila Giorgi.
And other winners on Wednesday were Klara Koukalova, Shuai Zhang and Aleksandra Wozniak.