The president spoke after a meeting with General David Petraeus, who handed command over the Multi-National Force in Iraq, this week to Genera Ray Odierno.
"One objective of these extremists is to kill, to try to cause the United States to lose our nerve and to withdraw from regions of the world," Mr Bush said.
The embassy assault was the deadliest attack on a compound that has been targeted four times in recent years by bombings, mortars and shootings.
Yemen, the ancestral homeland of
Osama bin Laden, has struggled to put down al Qaeda-linked Islamic militants, often to the frustration of US counter-terrorism officials
"Our message is that we want to help governments survive the extremists," Mr Bush said.
"We want people to lead normal life. We want mothers to be able to raise their sons and daughters in a peaceful environment so they can realise the hopes and dreams of a better world."
State Department spokesman Sean McCormack said one US embassy guard from Yemen was killed, along with several Yemeni security officials.
The assault bears "all the hallmarks of an al Qaeda attack," Mr McCormack said.
The
Yemeni government joined the US-led war against terrorism in the wake of 9/11.
The government of the poor Arab country has also been fighting Shi'ite rebels in the northern province of Saada since 2004 and faced protests against unemployment and inflation.
