Hyde fired the Bees in front in the 81st minute with a close-range finish from a Luke Gambin cross however the fairytale ending at Underhill looked like turning into a nightmare moments later.
Tom Flanagan fouled Wycombe midfielder Bruno Andrade in the box but Joel Grant's spot kick was pushed wide by Stack as Barnet held on to move out of the npower League Two drop zone.
After kick-off was delayed by 15 minutes, Barnet had the first two chances of the match with Hyde poking wide before seeing a header saved by Chairboys keeper Matt Ingram.
Curtis Weston almost put the Bees ahead just before the break with a powerful 25-yard effort, while player/manager Edgar Davids could not find the target when well placed.
Keanu Marsh-Brown then smashed over and substitute Gambin was denied by Ingram before Hyde and Stack combined late on with Barnet now a point above the drop zone with one match left.