The two were picked out by the voting public after being praised heavily by judges Sharleen Spiteri, Jamie Cullum and Dizzee Rascal.
Double act Pepper & Piano wowed the panel with a second rendition of their own track 'You Took My Heart', while County Clare musician Daithi impressed with his loop station and new composition 'Carroroe'.
Both acts will now appear at the final at Wembley, where they will compete for the £100,000 prize fund.
Welsh grime collective Flow Dem, electro-pop eccentrics Legion Of Many and Brit school band Chakula Soul were eliminated from the competition after failing to pick up enough public support.
Must Be The Music's second live semi-final airs next Sunday at 7pm on Sky1.