The British No. 2 arrived in Birmingham fresh from her run to the semifinals of her home tournament in Manchester.
And Broady carried that grass-court form into the clash against the world No. 39, recovering from 3-0 down in the opening set to win 7-6(3), 6-0.
It is the fifth time in her career that the 27-year-old has beaten a top-50 player and is likely to earn her a clash with two-time Wimbledon champion Petra Kvitova.
The British No. 3 played an excellent second set to level the match but one break of serve in the decider proved decisive, with Svitolina going on to win 6-2, 5-7, 6-3.
The Ukrainian is the highest-ranked player in the tournament after Angelique Kerber pulled out with a hamstring problem — one of a rash of withdrawals to hit the event.
Watson was too error-strewn in the opening set to put any real pressure on her opponent but recovered well in the second.
After saving break points at 1-2 and 2-3, Watson found more aggression on her shots and broke the Svitolina serve to lead 6-5.
She served out the set confidently, finishing with a forehand winner down the line.