This paper aims to address the security challenges on physical unclonable functions (PUFs) raised by modeling attacks and denial of service (DoS) attacks. We develop a hardware isolation-based secure architecture extension, namely PUFSec, to protect the target PUF from security compromises without modifying the internal PUF design. PUFSec achieves the security protection by physically isolating the PUF hardware and data from the attack surfaces accessible by the adversaries. Furthermore, we deploy strictly enforced security policies within PUFSec, which authenticate the incoming PUF challenges and prevent attackers from collecting sufficient PUF responses to issue modeling attacks or interfering with the PUF workflow to launch DoS attacks. We implement our PUFSec framework on a Xilinx SoC equipped with ARM processor. Our experimental results on the real hardware prove the enhanced security and the low performance and power overhead brought by PUFSec.