On-loan Southampton winger Elyounoussi is enjoying a new lease of life with the Glasgow giants and opened the scoring with a powerful header from James Forrest's cross before Callum McGregor tapped into an empty net from Odsonne Edouard's chip over Hibs goalkeeper Chris Maxwell.
Melker Hallberg briefly cut the Edinburgh side's deficit in half before Edouard was the creator again for Elyounoussi to smash home his seventh goal in seven games. Forrest and Elyounoussi then hit the woodwork early in the second half as Celtic threatened to run riot. But when the fourth goal did arrive, Hibs were masters of their own downfall as Maxwell flapped at a corner and Celtic captain Brown swept home the first of a rare double.
Hibs responded quickly through Florian Kamberi, but it was too little, too late with the pressure on manager Paul Heckingbottom ramped up once more with his side just a point off the bottom of the Scottish Premiership. Brown sealed a 30th straight domestic cup victory for the holders in stoppage time at the second attempt after Maxwell did well to block his first effort.
Celtic will now face old rivals Rangers or Hearts, who face off in the second semi-final on Sunday, for the first silverware of the Scottish season on December 8.