South Australia ended the opening day of play at 8/352, having recovered from 2/25 and 5/143 after being sent in to bat.
Cooper led the way but the veteran was unlucky to miss out on his 14th first class century after being bowled by Joel Paris on 99.
Along with wicketkeeper Nielsen (53), Cooper helped SA counterpunch after the lunch interval with an 80-run stand for the sixth wicket.
The tail also added invaluable runs late with Chadd Sayers (38*) and Daniel Worrall (19*) set to resume at the crease on Saturday.
Earlier, opener Jake Weatherald scored a composed 60, while Test batsman Travis Head added a handy 46 which featured seven fours.
Western Australia's seamers enjoyed the overcast conditions in the first half of the day utilising the swing-friendly Dukes ball, which has again been introduced for the second part of the Shield season.
Paris and David Moody both returned 3/70, while wicketkeeper Josh Inglis claimed three catches, as did Shaun Marsh at second slip.
The clash looms as a crucial one with the Redbacks within nine points of the second-ranked WA as the race for a finals berth begins to heat up.