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The referee stopped the contest at the Titanic Exhibition Centre in the fourth round after the outclassed Slavchev lifted the Belfast man off his feet.
Flyweight Barnes, 29, had been on top against the Bulgarian who had lost 24 of 33 previous bouts.
Light-flyweight Jamie Conlan beat Hungarian David Koos on points.
The Belfast boxer was the victor on a unanimous decision and now awaits a final eliminator for a world title fight.
Marco McCullough defeated Luis Ligo in his fight for the vacant WBO Intercontinental featherweight championship when his opponent failed to come off his stool for the fifth round.
Barnes's professional bow was much anticipated after a distinguished career in the amateur ranks.
"It was hard to pin him down," said Barnes after the strange conclusion. "I was trying too hard to impress. You didn't see the best of me."
Earlier, Belfast's Steven Ward, 26, marked his professional debut with a points win over 31-year-old Bulgarian Merdjidin Yuseeinov.
Ward, who won silver at the 2010 Commonwealth Games, produced a steady display and was declared the winner by 40 points to 36.
Tyrone McKenna won the Celtic super-lightweight belt by stopping Dublin opponent Sean Creagh.
Unbeaten McKenna, 26, was well on top when Creagh's corner retired their man before the sixth and final round.
Double Olympic bronze medallist Paddy Barnes won his pro debut fight in unusual circumstances when opponent Stefan Slavchev was disqualified.