The seventh seed advanced in straight sets, 6-3 6-3, and looked in impressive form with both her forehand and backhand as she forced 10 break points, converting four of them.
Dutchwoman Bertens could only break Konta once in reply and the British player is an 8/1 chance with Sky Bet for the US Open at the end of the month, the same price she is to land this title.
Konta will now await the winner of the second-round tie between Frenchwoman Alize Cornet and 11th seed Dominika Cibulkova.
World number two Simona Halep coasted into the third round with a straight-sets win over Taylor Townsend.
The Romanian, given a first-round bye, came through 6-4 6-1 in one hour and seven minutes against the American qualifier and is one of three 7/1 co-favourites for the title.
Garbine Muguruza is also a 7/1 shot having wasted no time in advancing to the third round with a 6-2 6-0 win over Brazilian qualifier Beatriz Haddad Maia, while Ekaterina Makarova was almost as dominant in beating Barbora Strycova 6-4 6-1 in their first-round tie.
Number nine seed Venus Williams booked her place in round two with a straight-sets 6-2 6-0 win over fellow American Alison Riske.
French Open champion Jelena Ostapenko, though, is out after the Latvian number 12 seed lost 6-4 6-2 to Aleksandra Krunic of Serbia.
Cibulkova, the 11th seeded Slovakian, needed three sets to defeat Ana Konjuh of Croatia, the world number 11 eventually coming through 5-7 7-5 6-2 following almost two-and-a-half hours.
Poland's number 10 seed Agnieszka Radwanska went down 6-4 6-4 to Julia Goerges of Germany, while 15th seeded Latvian Anastasija Sevastova came through 3-6 6-4 6-2 against Shuai Peng of China.
Australian qualifier Ashleigh Barty won 6-4 6-4 against American Varvara Lepchenko to set up a second-round date with Williams.
Related links
Sky Bet's latest WTA Tour prices
