Beth Mooney, Nicole Bolton and Ellyse Perry all struck half-centuries for the Southern Stars with Sune Luus taking five wickets as Australia was all out for 269 in 48.3 overs.
Jonassen, who finished with two for 40 from her 10 overs, said: “It was a nice hit out but we left a few runs out there. It was pleasing though with the ball to bowl them out for 210 in the end.
“We started a little bit off with the ball and strayed a bit with our lines early but the pleasing thing was how we brought it back in the middle overs and got crucial wickets at crucial times.
“We try and take it one game at a time and take each game in isolation, going into the semi-final that’ll be no different whoever we come up against.”
Haynes, in for the injured Meg Lanning, won the toss and elected to bat with Mooney and Nicole Bolton smashing their second century partnership of the tournament.
Bolton brought up her half-century first with Mooney following her to the landmark just two overs later.
Ayabonga Khaka eventually bowled Mooney with the score on 114 with Haynes dismissed shortly after for a duck followed by Bolton (79) and Elyse Villiani for a golden duck.
Perry rescued the innings with a fluent 55, her fifth consecutive half-century in the World Cup, but Australia lost their final six wickets for just 34 runs with Luus picking up a five-wicket haul.
Dane Van Niekerk and Karizanne Kapp both grabbed a pair of wickets as Australia was dismissed for 269 in the penultimate over.
After losing Lizelle Lee early, Wolvaardt and Trisha Chetty built steadily with the former bringing up her half-century with a push to deep cover off the bowling of Kristen Beams.
Captain Haynes then dismissed Mignon Du Preez (20) and Wolvaardt in identical fashion in consecutive overs as both batters were caught at deep midwicket by Ashleigh Gardner.
Kapp (19), Van Niekerk (1), Luus (6) and Khaka (0) all fell in the space of three overs as South Africa collapsed from 162-4 to 165-8 in the latter stages and the Proteas were eventually all out in the final over.
Luus said: “It’s always great to get five wickets in a World Cup, especially against a team like Australia, a five-for is really special no matter how it comes.
“It was a great performance by the team unfortunately we didn’t win, we’ll keep that for the semi-finals. I think it’s a first for us (reaching the semis) and the girls are very excited to go back to Bristol.
“We’re very excited to play England, hopefully we’ll get the win to get through to the final.”
Scores in brief:
Australia beat South Africa by 59 runs at The County Ground, Taunton
Australia 269 all out in 48.3 overs (Nicole Bolton 79, Ellyse Perry 55, Beth Mooney 53; Sune Luus 5-67, Marizanne Kapp 2-26, Dane Van Niekerk 2-41)
South Africa 210 all out in 50 overs (Laura Wolvaardt 71, Trisha Chetty 37, Mignon Du Preez 20; Rachael Haynes 2-12, Jess Jonassen 2-40, Ellyse Perry 2-47)
Player of the Match – Ellyse Perry (Australia)
Next Matches – England v South Africa, The County Ground, Bristol 18 July, Australia v India, The County Ground, Bristol, 20 July