Evans, who marked a major milestone in his career by making the semi-finals of ATP Zagreb last week, got off to a perfect start against the world number 91 as he took the opening set, but went on to lose 3-6 7-6(4) 6-4.
The British number two saved five break points as he claimed the opening set, and a further eight as the second set went to a tie-break - though he missed six break point opportunities of his own before losing that tie-breaker.
The final set was just as tightly fought, with Stakhovsky breaking early on and Evans breaking back, only for the Ukrainian to break once more to claim victory and earn a spot in the last 16.
There was more disappointment for Britain when world number 322 Kyle Edmund was also beaten 7-5 6-1 by Tobias Kamke.
World top 10 stars Richard Gasquet and Jo-Wilfried Tsonga are both playing the event in front of their home fans, with Gasquet to start his tournament in the last-16 against Blaz Kavcic.
Results from the Marseille Open Men's Singles Round 1 matches
Tobias Kamke (Germany) beat Kyle Edmund (Britain) 7-5 6-1
5-Ivan Dodig (Croatia) beat Albano Olivetti (France) 7-5 6-2
6-Edouard Roger-Vasselin (France) beat Ante Pavic (Croatia) 3-6 7-6(5) 7-5
Sergiy Stakhovsky (Ukraine) beat Daniel Evans (Britain) 3-6 7-6(4) 6-4
Nikolay Davydenko (Russia) beat Igor Sijsling (Netherlands) 6-4 6-4
Results from the Delray Beach International Men's Singles Round 1 matches
Ivo Karlovic (Croatia) beat Samuel Groth (Australia) 7-6(4) 6-3
1-Tommy Haas (Germany) beat Wayne Odesnik (U.S.) 6-2 6-1
Ryan Harrison (U.S.) beat Lu Yen-Hsun (Taiwan) 4-6 1-0 (Lu Y H retired)
Marinko Matosevic (Australia) beat David Goffin (Belgium) 6-2 6-4
Steve Johnson (U.S.) beat Mikhail Kukushkin (Kazakhstan) 6-4 6-3
8-Lleyton Hewitt (Australia) beat Bradley Klahn (U.S.) 6-3 6-1
Teymuraz Gabashvili (Russia) beat Matthew Ebden (Australia) 2-6 6-3 6-2
4-Kevin Anderson (South Africa) beat Tim Smyczek (U.S.) 4-6 6-1 6-3
Marcos Baghdatis (Cyprus) beat Jiri Vesely (Czech Republic) 3-6 6-2 6-1
Dudi Sela (Israel) beat Alex Bogomolov Jr (Russia) 6-0 6-2
Results from the Rio de Janeiro Open Men's Singles Round 1 matches
7-Juan Monaco (Argentina) beat Horacio Zeballos (Argentina) 6-3 6-1
4-Tommy Robredo (Spain) beat Pablo Carreno (Spain) 7-6(5) 6-3
Joao Sousa (Portugal) beat 6-Marcel Granollers (Spain) 2-6 6-3 7-5
Alexandr Dolgopolov (Ukraine) beat 5-Nicolas Almagro (Spain) 2-6 6-3 6-0
Albert Ramos (Spain) beat Leonardo Mayer (Argentina) 6-2 6-0
8-Pablo Andujar (Spain) beat Julian Reister (Germany) 6-4 6-3