Chariot duels in battles existed in the Spring and Autumn Period. In the Battle of Chengpu (632BC), it was recorded in the Zuo Zhuang that Ziyu of Chu sent a challenge to Duke Wen of Jin.

子玉使鬥勃請戰，曰，請與君之士戲，君馮軾而觀之，得臣與寓目焉。

Ziyu sent Dou Bo to request a duel, saying: I would like to cross swords with one of your knights, you can watch from you chariot and I will watch with you [from my chariot].

However, as the way warfare was conducted changed throughout the Spring and Autumn and Warring States period, duels began to die out as well.

Military aside, Han Feizi did record there were bands of "knights" who traveled around dueling. These people weren't part of any military nobility though, they were more like adventurous brigands who dueled for sport.

If you mean one-on-one duels, they were few and far between. /u/lordtiandao has given a great example of one during Spring Autumn, when honour still mean a little in battles.

However, if you mean slaying the commander during the battle proper which resulted in the battle being won, then that was more common.

During the Warring States, the Battle of Changping was won by Qin when Bai Qi 白起 tricked Zhao Kuo 赵括 into taking the initiative by attacking Qin fortifications. The Zhao army was then encircled and trapped on a hill, and when Zhao Kuo tried breaking the encirclement, he was shot to death by arrows. The death of their commander led to the Zhao's army surrender, and thus the battle was won.

Likewise during the Battle of Jiangxia during the Three Kingdoms, Huang Zu 黃祖 and his troops had held the upper hand against Sun Quan 孙权's army because he was upstream from the enemy. By sinking two warships (meng chong) to block of the river, he ensured that Sun Quan's troops were showered with arrows whenever they approached. Then Dong Xi 董袭 and Ling Tong 凌统 cut off the ropes tying the ships together and Sun Quan's troops broke through. Huang Zu's commander Chen Jiu 陈就 was then personally killed by Lv Meng 吕蒙, which caused the collapse of troop morale and the battle was lost. In the banquet that followed, Sun Quan declared that the battle was won due to Chen Jiu's death, and rewarded Lv Meng greatly for the deed.

Clarification, he's talking about Kingdom. Kingudamu is the literal transliteration of the Japanese name.

Everyone is mentioning duels but in the Kingdom manga the commander is usually killed in battle by an enemy who is aiming to kill them. The method of killing the commander is varied but usually involves a unit pushing through his lines and killing him.

The result is that without leadership the army would flee, become demoralised or not be able to react properly without a leader.

Killing the enemy commander is often a significant part of the plan to defeat the opposing army. The plan boils down to kill the commander in battle, his army falls apart, we win.

I imagine OP wants to know was killing the enemy commander in battle a serious strategy of the period also would killing the commander cause the army to rout.

Source: read the manga.

Not having read the manga, do you have specific cases in mind?