Quasi-periodic fast-propagating (QFP) magnetosonic waves and extreme-ultraviolet (EUV) waves were proposed to be driven by solar flares and coronal mass ejections (CMEs), respectively. In this Letter, we present a detailed analysis of an interesting event in which we find that both QFP magnetosonic waves and EUV waves are excited simultaneously in one solar eruption event. The co-existence of the two wave phenomena offers an excellent opportunity to explore their driving mechanisms. The QFP waves propagate in a funnel-like loop system with a speed of 682-837 km s(-1) and a lifetime of 2 minutes. On the contrary, the EUV waves, which present a faster component and a slower component, propagate in a wide angular extent, experiencing reflection and refraction across a magnetic quasi-separatrix layer. The faster component of the EUV waves travels with a speed of 412-1287 km s(-1), whereas the slower component travels with a speed of 246-390 km s(-1). The lifetime of the EUV waves is similar to 15 minutes. It is revealed that the faster component of the EUV waves is cospatial with the first wavefront of the QFP wave train. The QFP waves have a period of about 45 +/- 5 s, which is absent in the associated flares. All of these results imply that QFP waves can also be excited by mass ejections, including CMEs or jets.