"We recognise that things didn't work out, we are responsible for that and we are more disappointed than anyone else," the Uruguayan told the Canal N cable news channel.
"I said all along that if we didn't qualify, we couldn't continue and that's what happened. They offered us the chance to carry on and we said we would keep our word, we couldn't continue having not qualified."
The 68-year-old has previously coached clubs in Paraguay, Chile and Peru, winning domestic titles in all three countries, as well as Greece and Mexico.
He led Paraguay through 2002 World Cup qualifying but was then sacked and replaced by Italian Cesare Maldini for the finals.
During his spell in Greece, where he coached Ionikos, Panathinaikos and Iraklis, he was known for his so-called "tsuku tsuku" football which involved grinding out low-scoring wins.
Peru, who have not qualified for a World Cup since 1982, finished seventh in the nine-team South American qualifying group with 15 points from 16 games. They managed only one point in eight games away from home.
Their most impressive performance under Markarian was at the Copa America in 2011 where they reached the semi-finals.
- Sports & Recreation