The partnership - the highest for any team in this season's tournament - gave South Australia the perfect base to chase down Western Australia's competitive tally of 5-285.
QUICK SINGLE: Warner's ton can't save the Blues
Klinger further pressed his claims for Australian selection and he is now the competition's top run-scorer, having accumulated 338 runs at an average of 112 with two 50s and a century.
Earlier, former Australian Test opener Simon Katich again proved his quality with a sparkling rapid-fire innings of 66 for WA.
Having top-scored for the Western Warriors with 67 in Tuesday's defeat against Victoria, Katich again led the way for his side.
But it was Katich that again was the pick of the batsmen with his 66 coming off just 39 deliveries, in a chanceless innings that included five towering sixes.
QUICK SINGLE: RYOBI One Day Cup schedule and table
The 38-year-old veteran of 56 Test matches, discarded by Australian selectors in 2011, recently completed a stellar season in English country cricket scoring over 1,000 runs at an average of 73.
Peter George claimed the wicket of Katich, and was the pick of the South Australian bowlers with 2-41 off his ten overs.
The win keeps the Redbacks' slim finals hopes alive but ends Western Australia's one-day Cup ambitions.
Cooper said the conditions presented a massive challenge for his side.
"This wind was probably something that none of us had ever played in before," he said.
"It was quite tough, especially for bowlers running into it."
"It was good to be able to help the team get in a position where we could get over the line for our first win for the season."
WA captain Michael Beer was dissatisfied with his team's bowling effort.
"It was not good enough," he said.
"We did enough with the bat, even though we could have got a bit more. If we had have bowled well enough we would have won the game, but that didn't happen.
"Bowling at the death is a big one for us (to improve on). Leaking runs in that phase is what is hurting us."