The Carolingian Empire
As the Romans borrowed extensively from the Hellenistic culture, the Germans borrowed from the Romans. In this way it could be said that the Greeks and the Romans were the ones who left an imprint on Western culture, but the Germans, as the ones who exulted and preserved ancient Roman civilisation (such as the texts of Virgil and Tacitus), are responsible for the mark left upon us by the older civilisations.
The Germanic tribes had started off pagan, yet once converted, they were at least partly responsible for the deep roots Christianity took in Europe. England, as mentioned, began sending out its own missionaries to continental Europe, and Charlemagne fought and ruled as a Christian king. The close alliance of the Church and State that emerged in Europe could also be somewhat attributed to the Germanic kingdoms. Charlemagne tied the governance of his kingdom not to Frankish traditions, but instead to the hierarchical and common religion of Roman Catholicism. This intertwining of the secular and the religious left its indelible mark on European society, and it is reasonable to take it as one of the factors that set into motion the Reformation quite a few centuries later.
In addition to the adaptation of Roman culture, the Germanic peoples also brought to Europe their own novel influence, for example in the area of military organisation. The Germanic peoples, with their clans and comitatus, were inherently warriors. Their heavy cavalry mowed down the Roman legionnaires, and the powerful human-horse unit is a familiar symbol to us in medieval tales of jousting knights. (It is also an interesting aside that some attribute the Huns, the other barbarian group that swept across Europe temporarily, with the introduction of stirrups – a seemingly small but important innovation to mounted soldiers to the West.)
Thus, the Germanic domination over Europe was a slow, creeping process, which culminated in the integration of influences that were Roman, German, and Christian.