This paper reports on high-quality-factor (Q) acoustic-wave resonator (AWR)-based bandstop filters (BSFs) with continuously-variable stopband bandwidth (BW). They are based on mixed-technology resonant branches that are shaped by one acoustic-wave-lumped-element resonator (AWLR) and two lumped-element (LE) impedance inverters that are in-parallel cascaded to an all-pass-type network. Despite the use of AWRs, transfer functions with fractional bandwidths (FBWs) that are wider than the electromechanical coupling coefficient (kt2) of the AWRs can be created and controlled through the integration of variable LE capacitors that preserve the AWR's high-Q characteristics. A coupling-matrix-based design approach that facilitates the prediction of high- and low-frequency spurious modes — typically present in a realistic AWR filter response — already at the synthesis level is presented for the first time for AWR-based BSFs. An experimental BSF prototype constructed with commercially-available surface acoustic wave (SAW) resonators was built and tested at 418 MHz for proof-of-concept purposes. It exhibits tunable BW between 0.18–0.71 MHz (0.55–2.2kt2), passband insertion loss (IL) between 0.55–2 dB and maximum stopband attenuation between 25–40 dB (i.e., Q>8,000).