The Clarets squeezed through to the third round 4-3 on spot-kicks.
The Wales international had earlier produced fine stops from Jeff Hendrick, Dwight Gayle and Javier Manquillo to ensure the game finished 0-0 after 90 minutes in front of a crowd of 30,082 at St James’ Park.
In the process, he left the Magpies still awaiting a first victory of the season, although they will be kicking themselves after passing up a series of opportunities to win the tie.
Former Clarets midfielder Hendrick’s 10th-minute shot, which flew just wide, was as close as either side came to an early breakthrough.
The visitors started to impose themselves as the half wore on but created little of note. Jay Rodriguez’s 23rd-minute volley, which looped high over Freddie Woodman’s goal, represented their most telling effort.
With Sean Longstaff starting to bite in midfield and Ryan Fraser and Jamal Lewis down the left benefiting from a series of deft touches from striker Joelinton to deliver crosses into the box, the Magpies finished the first half strongly.
They should have been ahead within four minutes of the restart when Hendrick fired straight at Hennessey from six yards after Joelinton had scuffed Lewis’ cross into his path.
Gayle was desperately unfortunate not to open the scoring with 55 minutes gone when he out-stripped Nathan Collins and saw his left-foot shot come back off the underside of the crossbar.
Gayle and full-back Javier Manquillo were both denied by Hennessey as the home side turned the screw, but as time ran down, the prospect of penalties became inevitable.
And although Woodman saved from Josh Brownhill, Hennessey’s heroics allowed Charlie Taylor to score the winning spot-kick as Sean Dyche’s men prevailed 4-3.