The Saudi statement said it was responding to an incursion into its territory and attacks on its citizens.
Houthi fighters had crossed the border on Tuesday, killing a Saudi border guard and wounding 11 others.
A Saudi government adviser said jets had bombed Houthi camps inside Yemen on Wednesday and Thursday in response to the attack.
"We took back a small piece of territory and hit their camps around Saada," he said.
Late on Thursday, a US official said Washington was "concerned by the expansion of the conflict along the Saudi-Yemeni border."
"It's our view that there can be no long-term military solution to the conflict between the Yemeni government and the rebels," Ian Kelly, a state department spokesman, said.
"We call on all parties to the conflict to make every effort to protect civilian populations and limit damage to civilian infrastructure."
A five-year-old conflict between the Houthis seeking greater autonomy and the Yemeni government has intensified since August, with both sides increasing attacks.
Thousands of people have died and tens of thousands have been displaced due to the conflict, according to the Red Cross and the UN.
The Houthi fighters say Saudi Arabia has been aiding the Yemeni offensive against them and said their cross-border raid was an "act of revenge".