The Australian took 4-6 to help skittle Worcestershire for 100 in a five-wicket win at the Ageas Bowl, the Hawks now progressing to face Essex in the last four.
Ellis's astonishing figures and two wickets each for Chris Wood and John Turner maintained Hampshire's hopes of becoming the first team to win back-to-back Blast titles.
The Rapids had an enterprising 70-run stand between Usama Mir and Ed Pollock, but it was bookended by losing 6-29 and then losing the last four wickets for one run - Ellis destroying the tail with three wickets in four balls.
While Hampshire stayed on track for a third success in the competition, defeat for the Rapids resulted in them slipping to their sixth quarter-final exit in eight appearances.
Inaugural tournament champions Surrey sealed their place on Finals Day with a 13-run win over Lancashire in a high-scoring thriller at Emirates Old Trafford.
The 2003 winners defended a target of 188 to inflict a first home T20 defeat on Lancashire in 22 games dating back to late 2020 as the hosts slipped from 2-124 in the 14th over of their chase to finish on 6-174.
Surrey captain and England white-ball seamer Chris Jordan took 2-27 from four overs after his side finished on 5-187, with opener Laurie Evans hitting 70 off 41 balls and Jason Roy reaching 50 off 34.
Somerset lie in wait for Surrey in the semi-finals at Edgbaston a week on Saturday after a nail-biting five-wicket win against Nottinghamshire at Taunton.
After Nottinghamshire had posted 6-157 - Matthew Montgomery top-scored with 51 off 38 balls - Lewis Gregory (57 not out) and Ben Green (35 not out) put together an unbroken partnership of 96 to see their side home with three balls to spare.
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