Callum McGregor and Scott Sinclair fired Celtic to a 2-0 win over Old Firm rival Rangers 2-0 that keeps its treble hopes intact after reaching the Scottish Cup final.

Brendan Rodgers's men have yet to be beaten in domestic competition this season and there was little sign of that changing when McGregor's clever finish put the Bhoys ahead in the semi-final contest at Hampden Park.

Sinclair, who has 20 league goals to his name this term, then blasted a 51st-minute penalty past Wes Foderingham, and the only sour note for a Celtic side that dominated the derby was a first-half injury to Moussa Dembele.

Celtic, which has already sewn up its defence of the Scottish Premiership and won the League Cup, can now win a first domestic treble since 2001 by beating Aberdeen in the final on 28 May (AEST).

After a pulsating start, Celtic had the lead in the 11th minute when Dembele brilliantly teed up McGregor, who side-footed a neat first-time finish past Foderingham.

Celtic completely dominated the opening half, but was dealt a blow when Dembele pulled up with a hamstring injury and was replaced by Leigh Griffiths.

Griffiths had an impact as he was wiped out by a sliding James Tavernier in the penalty area, affording Sinclair the chance from 12 yards, which he took confidently.

Kenny Miller saw a downward header from Tavernier's cross well saved low to his right by Craig Gordon, while the former Celtic striker drew a smart stop from the Scotland goalkeeper's feet.

But Rangers, which lost for the first time under boss Pedro Caixinha, never really looked like mounting a comeback and Celtic, beaten by the Gers at the same stage last season, almost had a third when Tom Rogic crashed a late effort against the post.