Yes and No.
You may need to specify what you're talking about - virtually all weapons in the civil war used cartridges, be them paper cartridges for rifle-muskets and percussion carbines, or metallic cartridges for rimfire or centerfire-shooting breech-loaders. There are multiple significant instances of repeating rifles, always Union, facing confederates with the typical smoothbore and rifled percussion muskets and decimating them, from small-but-famous battles like Hoover's Gap in 1863, where a tiny number of a few hundred held off several thousands and inflicted many casualties, to the battle of Franklin, one of the most significant battles of the war, where nearly 10,000 mounted infantry and cavalry with the Spencer repeating rifle helped to turn the tide against the Confederates.
For reasons like these the other original response here is simply wrong, as the Spencer rifle & carbine for instance numbered at 106,667 total rifles produced from 1861-66, with the company producing them going kaputt due to a lack of demand with the market so cluttered with surplus, and even the most conservative estimates of the amount of the previous figure are at around 48-55,000 carbines alone produced, along with the definitely finished orders of 12,000 rifles. Keep in mind all sources believe that the number of Spencers bought personally exceeds the 106,667 figure indefinitely.
In addition, to name the major battles - Chattanooga, Nashville, Franklin, Five Forks, Selma, Chickamauga, Gettysburg, Shelbyville, Trevilian Station - not all saw large or decisive use, but most of these significant and infamous battles I have mentioned did indeed. It is worth noting that the Spencer is the most common breech loader of the war. This contradicts the information of other answers, which repeat a common misconception, severely underplaying the usage of any breech-loaders.
The Tarpley carbine, Morse and Richmond variant of the sharps were Confederate breech-loaders - which did indeed exist - and there are numerous photographs that exist of Confederates with the Maynard carbine, which proved popular with them, its metallic cartridge being reloadable up to 100 or more times. They did not have breech loaders at all in the same numbers as the Union, though, often making use of captured supplies or guns. They certain had units that used breech loading weapons, mainly cavalry. The Spencer rifle for the Union did play key roles in a number of the most major battles of the war and this is often understated, as we have seen, and the total number of breech loaders in the civil war is believed, from capandball.com using actual figures and data to be 427,000+, which in a war of approximately 3 and a half million men at arms is a high figure.
In short, I think some cavalry skirmishes did take place between breech-loading users on both sides in the civil war, major battles and decisive usage was however usually one-sided. No repeaters were used by the US Army in the battle of Little Bighorn, use was by the Indians who were very successful with them. The Spencer and Henry always worked well, there were no limited or marginalizing technical issues with them either. Wherever you heard this was misleading, as to begin with Custer and his men were using the Springfield trapdoor carbines at the Little Bighorn.