Precise regulation of span tension in web-handling machines is critical to ensure product quality. Elevated tension causes web fracture during operation; depressed or highly variable tension leads to irregular and unacceptable spool. Web-handling machines typically employ costly load cell sensors to measure span tension. This work investigates the use of nonlinear estimators or observers to provide cost-effective span tension estimates from system measurements. After presenting the dynamic equations of a web-handling machine, their structure is exploited to design a reduced-order, nonlinear observer with linear time-varying error dynamics. Exponential stability of the error dynamics is demonstrated. Performance of the estimator is experimentally compared with two competing designs on a real web-handling machine.