Sepsis is an organ dysfunction caused by a deregulated immune response to an infection. Early sepsis prediction and identification allow for timely intervention, leading to improved clinical outcomes. Clinical calculators (e.g., the six-organ dysfunction assessment of SOFA) play a vital role in sepsis identification within clinicians' workflow, providing evidence-based risk assessments essential for sepsis diagnosis. However, artificial intelligence (AI) sepsis prediction models typically generate a single sepsis risk score without incorporating clinical calculators for assessing organ dysfunctions, making the models less convincing and transparent to clinicians. To bridge the gap, we propose to mimic clinicians' workflow with a novel framework SepsisCalc to integrate clinical calculators into the predictive model, yielding a clinically transparent and precise model for utilization in clinical settings. Practically, clinical calculators usually combine information from multiple component variables in Electronic Health Records (EHR), and might not be applicable when the variables are (partially) missing. We mitigate this issue by representing EHRs as temporal graphs and integrating a learning module to dynamically add the accurately estimated calculator to the graphs. Experimental results on real-world datasets show that the proposed model outperforms state-of-the-art methods on sepsis prediction tasks. Moreover, we developed a system to identify organ dysfunctions and potential sepsis risks, providing a human-AI interaction tool for deployment, which can help clinicians understand the prediction outputs and prepare timely interventions for the corresponding dysfunctions, paving the way for actionable clinical decision-making support for early intervention.