Exploration in environments with sparse rewards remains a challenging problem in Deep Reinforcement Learning (DRL). For the off-policy method, it usually needs a large number of training samples. With the growing dimensions of state and action space, this method becomes more and more sample-inefficient. In this paper, we propose a novel fast exploration method for off-policy reinforcement learning, called Density-based Intrinsic Motivation and Self-adaptive Action Noise (DIMSAN). Our main contribution is twofold: (1) We propose a Density-based Intrinsic Motivation (DIM) method. It introduces a new intrinsic-reward generation mechanism based on samples' density estimation during experience replay and encourages the agent to seek novel and unfamiliar states. (2) We propose a Self-adaptive Action Noise (SAN) to deal with the exploration-exploitation tradeoffs, which could automatically change the exploration step through adding adaptive action space noise. The synergy between DIM and SAN could guide the agent to search the state and action space with high efficiency. We evaluate our method on the benchmark manipulation tasks and the designed challenging ones. Empirical results show that our method outperforms the existing methods in terms of convergence speed and sample efficiency, especially in challenging tasks.