Leclerc clocked a best lap in one minute and 11.355 seconds to beat Piastri by 0.038s, a margin that suggests the champion team are likely to shine again on Saturday.
Djokovic will start his bid for a fourth French Open triumph and record-breaking 25th Grand Slam singles title when he faces American Mackenzie McDonald in the first round.
Zimbabwe, playing first Test in England for 22 years, were dominated by England's batters but had reached the relative comfort of 187-3 in reply on a flat pitch.