Regis Chakabva hit 42 off 22 balls not out for Zimbabwe but they came late as Afghanistan bowlers made some early breakthroughs to take the target beyond their reach.
Zadran smashed 69 off 30 balls, with five fours and six sixes, and shared 107 runs in a fifth-wicket stand with Mohammad Nabi.
The Zouks have announced a number changes to their roster for the upcoming season.
Afghanistan, playing in only their second World Cup, beat Sri Lanka soundly in last year's Asia Cup in the United Arab Emirates, but for the second match in a row this World Cup their batting let them down.
Chasing a revised target of 187 after the rain interrupted the proceedings, Afghanistan were all out for 152.
Afghanistan's total of 207 all out, made following captain Gulbadin Naib's surprising decision to bat first at Bristol on Saturday, never looked like troubling.
Australia will be firm favourites to knock off the runs in the fourth match of a tournament characterised by one-sided contests so far.
Andrew Balbirnie hit a career-best unbeaten 145 to guide Ireland to a series-levelling four-wicket win against Afghanistan in the third ODI Tuesday.
After suffering an initial collapse, Afghan and Zadran showed a lot of character to pull off a rescue effort with a 117-run partnership.
Afghanistan outplayed Ireland by five wickets to win the first ODI and take 1-0 lead in five-match series.