Set 272 for victory from 90 overs on Friday at the WACA Ground, WA went from a comfortable 2-155 to a shaky 9-249 as Queensland's attack put the squeeze on.
But Rocchiccioli (24no off 37 balls) and No.11 Cameron Gannon (10no off 13 balls) showed nerves of steel in an unbeaten 23-run final-wicket stand to guide WA to their first win of the season with 7.3 overs remaining.
Whiteman, in his 100th Shield match, was the rock at the top of the order, with his knock of 81 off 170 balls helping set up victory.
But he had some luck along the way.
The 32-year-old was yet to score when he was dropped at second slip by substitute fielder Hugh Weibgen, who was on the field for the rested Matt Renshaw (knee).
Whiteman was dropped again on 47, this time in the deep by another substitute fielder - Benji Floros.
Cameron Bancroft (36), Hilton Cartwright (25) and Cameron Green (17) failed to go on with their starts, while Cooper Connolly was out for just two as Queensland worked their way back into the contest.
Xavier Bartlett (3-25) loomed as the biggest threat to WA's victory hopes, with the home side going from 2-155 to 4-159 when the paceman struck twice in the space of three overs.
Whiteman's luck finally ran out when he hooked Gurinder Sandhu to the deep and Angus Lovell took a tumbling catch, reducing WA to 5-180.
WA still needed 55 more runs for victory when Aaron Hardie fell for 12, and when Josh Inglis departed for 28, WA's tail was well and truly exposed.
But Rocchiccioli and Gannon stood up when counted, reviving WA's season after the Warriors began with two losses and a draw.
It wasn't only Queensland's dropped catches on Friday that proved costly.
The Bulls were cruising at 4-183 in their second innings late on Thursday before disaster struck.
Man of the match Rocchiccioli snared four quick wickets as the Bulls slumped to 8-187 by stumps.
Rocchiccioli (5-47) claimed another wicket on Friday morning to help dismiss Queensland for 203, leaving WA with 90 overs to chase down the victory target.
