The Air Bagan ATR 72 twin-engine turboprop, carrying 49 passengers from the central city of Mandalay, "veered slightly off the runway due to bad weather and heavy rain" on landing Friday evening, the airline said in a statement posted on Facebook.
It said that all passengers were "safely disembarked" and only one, a Buddhist monk, sustained a slight injury to his hand.
It was not immediately possible to reach the airline or Myanmar's Civil Aviation Department for further comment.
In December 2012 an Air Bagan plane crash-landed in thick fog and burst into flames in a field short of the runway at Heho airport – the gateway to the popular tourist destination of Inle Lake – in northern Shan state, killing a tour guide on board and a motorcyclist on the ground as well as injuring 11 people including foreigners.
The airline was launched in 2004 by business tycoon Tay Za, who is known for his close links to the former junta, as the first private carrier in Myanmar.
With a fleet comprising one ATR 42 and three ATR 72 aircraft, it serves 19 destinations in Myanmar and also runs a charter flight to the popular holiday spot of Chiang Mai, according to the airline's website.