Bartoli, the French number one, put herself and her fans through the wringer in a 7-6 (8) 4-6 7-5 victory.
The unorthodox 28-year-old, known for her eccentric on-court mannerisms including the tennis equivalent of shadow-boxing and swishing at invisible balls to practise her swings, was set and a break ahead before rain halted her progress.
On the resumption Bealrussian Govortsova, who would give Maria Sharapova a run for her money on the grunting meter, played inspired tennis and served for the match at 5-3 in the decider, only to squander two match points.
Bartoli, roared on by the crowd, battled back to break serve for a 6-5 lead but then tamely double-faulted on her first match point in a game that went on for more than 10 minutes.
Three more match points went begging before former Wimbledon runner-up Bartoli finally claimed victory.
"I managed to ramp up a gear and to show that I wasn't tired at all, that's very encouraging for the rest of the season," Bartoli said.
Play did not start at the allotted 11:00 local time and, even when the green covers were hauled off the damp red clay back at lunchtime, only 90 minutes were possible before another rain delay sent fans and players scuttling for shelter.
Eleven singles matches were cancelled and three failed to finish meaning the first round will now span four days, once again raising questions about the tournament's decision to play only a partial schedule on the opening Sunday.
"At the moment it's not under consideration, it's not something that's been talked about," a tournament spokesman said.
The rain could not save the tournament's oldest player, though, with 42-year-old Japanese Kimiko Date-Krumm succumbing to Australian ninth seed Samanatha Stosur.
9-Samantha Stosur (Australia) beat Kimiko Date-Krumm (Japan) 6-0 6-2
13-Marion Bartoli (France) beat Olga Govortsova (Belarus) 7-6(8) 4-6 7-5
16-Dominika Cibulkova (Slovakia) beat Lesia Tsurenko (Ukraine) 6-1 6-4
18-Jelena Jankovic (Serbia) beat Daniela Hantuchova (Slovakia) 6-4 7-6(7)
27-Yaroslava Shvedova (Kazakhstan) beat Coco Vandeweghe (U.S.) 6-0 3-6 6-2
31-Alize Cornet (France) beat Maria Joao Koehler (Portugal) 7-5 6-2
Mariana Duque (Colombia) beat Kristyna Pliskova (Czech Republic) 6-2 6-0
Kristina Mladenovic (France) beat Lauren Davis (U.S.) 6-0 7-5
Silvia Soler Espinosa (Spain) beat Irina Begu (Romania) 6-3 6-2
Marina Erakovic (New Zealand) beat Elena Baltacha (Britain) 6-3 6-0
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