In a televised speech, he said the troops were sent into the neighbouring country to defend its sovereignty against the Islamic Courts Union (UIC).
While Mr Meles was speaking, his troops were attacking the Islamist militias in four areas of southern Somalia.
The BBC's Amber Henshaw in Addis Ababa says there has been a significant escalation in the conflict.
The UIC, which has seized control of much of southern and central Somalia, says Ethiopian troops have been fighting alongside government forces for months.
Fighting between the transitional government and forces loyal to the UIC now stretches over 400km (250 miles), our correspondent says.
"As of today our defence forces have launched a counter-offensive, which is completely legal and proportional, on these anti-peace forces [the UIC]," Mr Zenawi said.
"We are not trying to set up a government for Somalia, nor do we have an intention to meddle in Somalian internal affairs. We have only been forced by the circumstances.
"We want to end this war urgently and we hope that Ethiopian people stand by the defence forces," he said.
Ethiopia says its forces have killed hundreds of Islamists, but Ibrahim Hassan Addow, foreign secretary for the UIC, told the BBC the Ethiopians were lying and that his forces were holding their own.
Air strikes and shelling have been reported in border areas. Eyewitnesses said Ethiopian fighters were bombarding the UIC-held town of Beledweyne near the border, on the fifth day of renewed fighting.
"The enemy of Allah has started bombing our civilians," Sheikh Hassan Derrow told AFP news agency.
A resident told AFP: "We see planes striking us and heavy fighting on the ground intensifying."
Ethiopia, however, denies that its jets have dropped bombs.
The Islamist group - which controls most of the south, including the capital, Mogadishu - on Saturday appealed for foreign fighters to join them in a "holy war" against Ethiopia.
The UN estimates that at least 8,000 Ethiopian troops may be in the country, while rival Eritrea is said to have deployed some 2,000 troops in support of the Islamic group.
However, Eritrean President Isaias Afewerki denied his country had troops deployed in Somalia.