Network Function Virtualization (NFV) seeks to replace hardware middleboxes with software-based Network Functions (NFs). NFV systems are seeing greater deployment in the cloud and at the edge. However, especially at the edge, there is a mismatch between the traditional focus on NFV throughput and the need to meet very low latency SLOs, as edge services inherently require low latency. Moreover, cloud-based NFV systems need to achieve such low latency while minimizing CPU core usage. We find that real-world traffic exhibits burstiness that causes latency spikes of up to 10s of milliseconds in existing NFV systems. To address this, we built NetBlaze, which achieves sub-millisecond p99 latency SLOs, even for adversarial traffic, using a novel multi-scale core-scaling strategy. NetBlaze makes traffic-to-core allocation decisions at rack, server, and core-spatial scales, and at increasingly finer timescales, to accommodate multi-timescale bursts. In comparison with state-of-the-art approaches, NetBlaze is the only one capable of achieving sub-millisecond p99 latency SLOs while using a comparable number of cores.