Longtime ruler Yahya Jammeh, who came to power in a 1994 coup, has refused to step down despite losing a disputed December 1 presidential election to Barrow, deepening a political crisis.
In a statement, Senegal's army said on Thursday that forces from ECOWAS, West Africa's regional bloc,  had begun strikes as part of an operation aimed at upholding the result of last month's vote.
Colonel Abdou Ndiaye did not specify the type of strikes, but said "significant" land, air and sea resources had been made available .
Earlier on Thursday, Barrow, who had recently sought shelter in Senegal, took the oath of office in a hastily-arranged ceremony at Gambia's embassy in the Senegalese capital, Dakar.
"This is a day no Gambian will ever forget in a lifetime," Barrow said in a speech immediately after being sworn in.
Not long after his inauguration, the United Nations Security Council unanimously adopted a resolution backing Barrow and called for a peaceful transer of power.
"The people of The Gambia spoke clearly at the elections in December. They chose Adama Barrow to be their president. Their voice now needs to be heard and their will needs to be heeded by just one man," Peter Wilson, the UK deputy ambassador to the UN, said.
Earlier this week, Jammeh had declared a national state of emergency, while the parliament extended his term in office by 90 days. He has not been heard from since his mandate expired at midnight.
At least 26,000 people have fled Gambia for Senegal since the start of the crisis fearing unrest, the UN's refugee agency UNHCR said on Wednesday, citing Senegalese government figures.
Hundreds of people gathered outside Gambia's embassy in Dakar