Rinke hit 14 fours and three sixes in 104 at just under a run-a-ball to be the rock upon which Zimbabwe built a total of 304 for four from their allocation of 50 overs on Tuesday.
Fast-medium bowler Tawanda Mupariwa then took three wickets for 30 runs from seven overs and left-arm spin bowler Keith Dabengwa snared three for 49 from eight overs, as the hosts were dismissed for 253 with six balls to spare.
Antigua and Barbuda looked on course for victory, when Leeward Islands captain and West Indies batsman Sylvester Joseph and opener Rolston Phillip launched a merciless assault on the Zimbabwe attack.
Joseph, eyeing a return to West Indies colours, hit the top score of 49 and Phillip made 42 in a second-wicket stand of 94 before Mupariwa and Dabengwa slowed the home team's progress with some purposeful bowling.
After the departure of Joseph and Phillip, there was no substance and stability from the rest of the batting as the Zimbabweans celebrated their second straight victory over the Antiguans.
Earlier, Rinke shared three productive partnerships on the hard, easy-paced Jolly Beach Resort Ground pitch.
He shared 100 with captain Terry Duffin for the first wicket, then added 59 for the second wicket with Chamunorwa Chibhabha, and put on 32 for the third wicket with Brendan Taylor before his dismissal.
Duffin scored 32, Chibhabha made 21, and Taylor clouted seven fours and five sixes in 80 to give the innings a timely boost in the latter stages.
The Zimbabweans now commence final preparations for the first two limited-overs internationals against West Indies this weekend at the Antigua Recreation Ground.
Zimbabwe: 304 for four off 50 overs (Piet Rinke 104, Ryan Higgins 80, Elton Chigumbura 36 not out, Terrence Duffin 32, Chamunorwa Chibhabha 21; Anthony Martin 2-51).
Antigua and Barbuda: 253 off 49 overs (Sylvester Joseph 49, Rolston Phillip 42, Justin Athanaze 35, Devon Thomas 29, Bertel Baltimore 23; Tawanda Mupariwa 3-30, Keith Dabengwa 3-49).