Astle belted the last three balls of the innings for six to finish 118 not out in New Zealand's imposing 50-over total of 276 for six.
Ramnaresh Sarwan and Runako Morton both made half-centuries for the West Indies, who produced their best batting performance of the series, but could still only muster 255.
''We're pretty happy, it was a good performance,'' New Zealand captain Stephen Fleming told New Zealand Press Association.
''We felt good defending that score. It did get quite close, but it confirmed we are on track in defending scores.'' Astle, who was twice dropped from the team earlier this summer, was at his brilliant best as he reached his 16th ODI hundred and his first against the West Indies when he took a single off the first ball of Chris Gayle's final over.
The 34-year-old got himself back on strike for the last three deliveries and promptly smacked them all over the boundary rope to finish unbeaten on 118 from 126 balls.
''Getting dropped and coming back and proving yourself is what it's all about,'' he said. ''It's part and parcel of international cricket...so I suppose on that front it's very nice.'' Hamish Marshall, recalled to the team for Jamie How, also chipped in with a timely 43 off 54 balls in a record 99-run partnership with Astle for the fifth wicket.
The West Indies, beaten in the first two games of the five-match series, had looked to have gained the upper hand when fast bowler Ian Bradshaw ripped through the top order.
The lively paceman dismissed Lou Vincent (25), Fleming (15) and Scott Styris for one as the home team crashed to 64-3 by the 15th over.
The hosts then slumped to 87-4 when Peter Fulton fell to Dwayne Smith for 21, bringing Marshall and Astle together, who set about putting their team firmly back in control of the match.
The pair broke the 10-year-old record, set by Fleming and Chris Cairns, for the highest fifth-wicket partnership by a New Zealand pair against the West Indies.
Sarwan made 65 off 78 balls and Morton struck 58 off 62 deliveries for the West Indies but the tourists were always struggling to get near New Zealand's total after a slow start.
Opener Daren Ganga was bowled by Shane Bond (3-47) for one and Gayle departed for 38 just when he was starting to look dangerous.
Morton steadied the innings and Sarwan and Bradshaw (37) provided some lusty hitting after super-sub Jeetan Patel destroyed the middle-order with three quick wickets but it was all too little, too late.
The fourth ODI is at Napier on Wednesday with the final game at Auckland next Saturday.