Customized Large Language Model (LLM) agents face a critical security threat from black-box instruction backdoors, where malicious behaviors are covertly injected through hidden system instructions. Although existing prompt-based defenses can often detect poisoned inputs, they generally fail to recover correct outputs once the backdoor is activated. In this paper, we first conduct a mechanistic analysis of LLM behavior under instruction backdoors and reveal two pivotal phenomena: (1) cognitive override, in which backdoor triggers dominate the reasoning process and suppress task-relevant context, and (2) abnormal semantic correlation, where triggers establish excessively strong semantic associations with attacker-specified target labels. Based on these insights, we propose a $\textbf{S}$oft $\textbf{L}$abel mechanism and key-extraction-guided CoT-based defense against $\textbf{I}$nstruction backdoors in A$\textbf{P}$Is (SLIP). To counteract the cognitive override, the key-extraction-guided Chain-of-Thought (KCOT) explicitly guides the model to extract task-relevant keywords and phrases rather than only considering the single trigger or overall text semantics. To neutralize the trigger's abnormal semantic correlation, the soft label mechanism (SLM) quantifies semantic correlations and employs statistical clustering to filter anomalous phrases before aggregating reliable keywords and phrases for prediction. Extensive experiments show that SLIP reduces the average attack success rate to 25.13$\%$, improves clean accuracy to 87.15$\%$, and outperforms state-of-the-art black-box defenses.