I think it's reasonably clear from the figure that indeed it did not reach escape velocity after launch, but rather when it passed Jupiter. Indeed, NASA say as much:
Consider Voyager 2, which toured the Jovian planets. The spacecraft was launched on a Type-II Hohmann transfer orbit to Jupiter. Had Jupiter not been there at the time of the spacecraft's arrival, the spacecraft would have fallen back toward the sun, and would have remained in elliptical orbit as long as no other forces acted upon it. Perihelion would have been at 1 AU, and aphelion at Jupiter's distance of about 5 AU.
And also (from NASA's caption to the figure):
Voyager 2 leaves Earth at about 36 km/s relative to the sun. Climbing out, it loses much of the initial velocity the launch vehicle provided.
(This link is in the caption of the figure you included in Wikipedia! You could probably have found it with a little clicking.)