Well, the history of stirrups is rather long and windy. First of all, it's important to understand that the use of saddles predates stirrups. A saddle is also more than just a really bulky piece of leather strapped to the horse's back- it's a finely crafted device designed to spread the weight of the rider out over the horse's back by use of the tree (which is like the frame of the saddle). Early saddle were glorified pads, but as riding developed, so did tack, which led to early saddles with wooden trees. Stirrups cannot exist without a proper tree because they would cause substantial pain to the horse (by putting a lot of narrow pressure on its back), and they wouldn't do anything to facility stability while standing. Really, you'd be better off suspending yourself with your knees and thighs than the irons on a treeless saddle.

Most historians thing that the first people to use something stirrup like were Indians, who had toe-loops attached to the girth. However, the effectiveness of that in combat is questionable.

Stirrups were first used to facilitate the mounting of a horse, and then would be ignored once the rider was astride. It's conceivable that somebody might have used the one while riding to help with the draw of a bow, but there isn't really any evidence to support that notion.

According to a few archeological digs in Chinese tombs, the earliest known use of paired stirrups (which would be used for riding) was somewhere around the 4th century AD, and they were found in a Xianbei tomb.

That said, similar stirrups where found all over northern China, Korea, and even in parts of Japan, so it's hard to say who had them first. But, regardless of that, we start seeing more depictions of armored cavalrymen and more heavily armed cavalry around 400-500AD. And, according to most records of battles, those with more heavy cavalry generally were successful in battle.

So, we don't know the exact date of the battles that first used stirrups, but we have a good idea about when they happened, more or less.

Sources:

The Stirrup and its effect on Chinese Military History, by Albert Dien. Medieval Technology and Social Change, by Lynn White