Striker Leigh Griffiths headed his side into an early lead when he latched on to Stefan Johansen’s superb cross towards the far post.
Kris Commons added the second towards the end of a first half Celtic totally dominated, finding the top corner with a sweet strike from the edge of the box.
Rangers saw more of the ball after the break but their endeavour did not make up for the gulf in quality separated by an entire division of the Scottish football league.
This was the first Old Firm derby since Rangers were liquidated in 2012 and it provided the same historic intensity as all those that have come before.
But Celtic, who lead the Scottish Premiership, showed their top-flight quality when Johansen spotted a clever run from Griffiths and picked him out at the far post to head past Steve Simonsen and send the green and white end of Hampden Park into raptures.
Griffiths almost headed in a second four minutes later when he was found unmarked again, this time by Mikael Lustig, but directed his effort wide of the far post.
Defender Virgil van Dijk and Johansen also had chances from inside the box but Commons made sure Celtic were in firm control at the break when he controlled a loose ball on the edge of the box and steered a vicious effort beyond Simonsen’s reach on 31 minutes.
Rangers increased their intensity in the second half but struggled to create clear-cut chances, with half-time substitute John Daly making a nuissance of himself up-front.
Tempers became stretched and frayed, with familiar scenes of pushing and shoving after harsh challenges in midfield becoming regular in the final 10 minutes.
But Celtic held on to reach the League Cup final for the 30th time in their history – and give themselves the chance of a first triumph in the competition since 2009.