The Odyssey was summoned to Orilla by the Asgard to receive a massive upgrade of all Asgard technology as well as receiving all the Asgard's knowledge. The ship received an Asgard computer core, presumably newly improved shields, and Asgard plasma beam weapons, among other upgrades. However, being able to detect the new Asgard upgrades through subspace, three Ori warships became aware of their location and emerged from hyperspace to attack. (SG1: "Unending")
The Odyssey ran for it being chased by three Ori warships while the Asgard self-destructed Orilla. After sustaining several direct hits, the Odyssey used its new Asgard plasma beam weapons and quickly penetrated the Ori ship's shields, destroying one Ori warship with minimal damage to the Odyssey while it escaped into hyperspace.
After briefly exiting hyperspace, due to an abnormality in the Asgard hyperdrive, they were immediately attacked by 2 Ori warships. The Odyssey sustained several direct hits before retreating back into hyperspace.
Faced with being destroyed by 2 Ori ships and with a hyperdrive failure eminent, the Odyssey exited hyperspace over P3X-474. After beaming the crew to the surface, with the exception of SG-1 and Major General Henry Landry, Odyssey engaged the Ori ships, firing on only one target at a time. The Odyssey destroyed one more ship before losing shields and taking heavy damage. Not being able to survive one more blast, Lt. Colonel Samantha Carter erected a time dilation field around the Odyssey, seconds before being impacted by an Ori energy beam weapon.
After spending 50 years in the time dilation field, Carter found a way to reverse the field to the time they activated it, and a way to quickly separate the hyperdrive controls from the Asgard computer core. Back to moments before the field was erected, the Odyssey was able to enter hyperspace, leaving the remaining Ori ship over P3X-474, and returned to Earth. (SG1: "Unending")
After the Odyssey arrived back at Earth, they quickly were able to duplicate most of the Asgard technology, such as Asgard plasma beam weapons and the Asgard computer core. These advancements were later installed aboard all other Earth battleships, which gave them a significant advantage over most space-faring races and ultimately helped the Atlantis expedition defeat the Asurans once and for all. (SG1: "Unending", SGA: "Be All My Sins Remember'd")