In 1248 William II, count of Holland and King of Germany started building a hall that's today known as the Ridderzaal. His son Count Floris V finished the construction in 1280. This is where the history of the Hague begins.
In the 14th century the Hague became the administrative capital of the county of Holland.
In the 15th and 16th century the Netherlands were under Burgundian, Habsburg and Spanish rule. The Netherlands were divided into small 'states' run by nobles and burgeoisie. These small states gathered in Brussels to discuss matters in the realm with their King a few times. This council became more and more powerful and got a lot of privileges.
When the Dutch Republic seceded from Spain the states needed a new place to gather. At first this was the city of Middelburg in the county of Zeeland. When this location became unsave they relocated to the Hague. The reason was that it was a village of no significance. It didn't had any city rights which meant that it wasn't self governing and didn't had any representation on the council. Because of the fact that the village had no political power and was therefore neutral ground made the states decided to gather there.
This was also the time known in the Netherlands as 'the Golden Age'. The Netherlands was one of the most powerful nations and the biggest trade power. The center of all the trade was the city of Amsterdam. Amsterdam would always be the most important city of the Netherlands.
In 1815 when Napoleon was defeated the Kingdom of the Netherlands came to be. As Amsterdam was the most important city King William I decided that it would become the capital (it was already the de facto capital). Brussels and the Hague became the seats of the government like they used to be. After Belgium seceded in 1830 from the Kingdom the remaining seat of the goverment , the Hague, became the permanent seat of the government.
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Small note: I suck at English grammar
this is something that has its roots in the 80 Years War. After the new republic declared its independence, not all parts of what's now the Netherlands were willing to go along with this.
Because of trade considerations, Amsterdam initially remained loyal to Spain until it was forced to change sides after it became isolated in 1578, so logically the new country couldn't pick it as its capital even though it was the biggest and most powerful city in Holland.
The first Free assembly of the rebels in 1572 specifically calls out Amsterdam's reluctance to join, and proposes to send out letters to other protestant trade cities to effect a blockade of the city:
With respect to [the interests of] Amsterdam in the Sound, His Highness might write to Denmark and the other Baltic towns at the earnest request of the States and towns of Holland to the effect that because of Amsterdam's great enmity and opposition to freedom and the well-being of the common fatherland [gemeen vaderlants], it should be refused passage and that they [ie. Baltic towns] should put into and trade with Enkhuizen, Hoorn and the other nearby towns devoted to us or, in the case of the Maas, with Dordrecht, where they will find the situations and arrangements as convenient as Amsterdam
A second element that played a role was the position of the Stadhouder in the Republic, and by extension the House of Orange since they provided all the Stadhouders.
Amsterdam had a large degree of independence, as did a lot of other cities. But there was always a concern that the Stadhouder would gain too much power and effectively become a king, and the regents of Amsterdam tended to be at the forefront when it came to calling for restrictions to the power of the house of Orange.
Likewise the concerns were often justified as Stadhouders often fought to extend their power, and in 1618 and 1650 actively sought to overthrow the Republic to start a monarchy. The Republic did away with the position of Stadhouder twice (the first time after 1650 and until 1672 ), but in times of crisis reappointed them.
During the Napoleonic Wars, Amsterdam became the official capital with also the government residing there. After the Kingdom was re-established in 1815, the government moved back to the Hague, but as a gesture of reconciliation from the House of Orange, Amsterdam remained the official capital.
For more information I can recommend Jonathan I. Israel's "The Dutch Republic. It's Rise, Greatness, and Fall 1477-1806"
Short answer: the seat of political power, nowadays known as het Binnenhof, has been in the Hague since 1446. The wikipedia entry of het Binnenhof can give you some more details on the locale itself. This is where the nobility and other notables gathered during the late middle ages, renaissance and early modernity.
Longer answer: Dutch political power has been fragmented and split over many players. Amsterdam was one such a player and they rose to prominence most notably during the 17th century. That's when Amsterdam became a power player, but the power was still split over many players. A brief overview of players during the 17th and 18th century: Stadholder, the provinces, the cities (each city was a seperate power, although they did band together when politically expedient) and various local notables - merchants, regenten and so on. Perhaps I should list the VOC and WIC here as well, but they were an extension of the politicians and merchants. You then also had a split between 'pro-monarchy' and 'pro-republic' parties, but those political parties fluctuated greatly over the years. The city of Amsterdam didn't become a stronghold of the Stadholder during these years nor was it the seat of the nobility. In that sense, it never became the center of politics or power - it simply was a strong player, but not the center of politics.
tl;dr: the Hague is the seat of political power, Amsterdam was simply a key player and eventually the 'cultural' capital of the Netherlands, but the seat of politics never moved there.
I'm a bit under the weather though, so if I made any glaring errors or am unclear in any way, do ask/correct me.