Large Language Models (LLMs) are becoming key components in various mobile operating systems, driving smart applications like interactive chatbots and personal assistants. While bringing enhanced intelligence to mobile ends, their deployment suffers from a set of performance challenges, especially the generation quality degradation and prolonged latency. Prior works have mainly relied on solutions of cloud offloading or on-device Small Language Models (SLMs). However, the former is usually limited by the communication bottleneck, and the latter sacrifices generation quality due to resource constraints. To mitigate these limitations, this paper proposes Synera, a device-cloud synergistic LLM serving system that applies an efficient SLM-LLM synergistic mechanism. Through empirical studies on LLM's unique computing characteristics, Synera identifies a set of underexplored optimization opportunities in device-cloud synergistic LLM inference, including offloading decisions, pipeline stalls, and batching bottlenecks. To translate them into enhanced performance, Synera introduces tailored designs of communication-efficient selective offloading, stall-free parallel inference, and scalable cloud batching. Extensive evaluations with real-world testbeds show that Synera enables 1.20-5.47x better generation quality against competitive baselines with on-par latency performance. Compared with existing cloud serving, Synera achieves 8.2-16.5% lower cloud serving cost on various benchmarks.