Wimborne, who are currently second in the Dorset top flight, stormed into the semi-finals thanks to a six-goal blitz in the first half.
Delighted Wimborne boss Phil Effick hailed an all-round team performance that saw five different players getting on to the scoresheet.
“The majority of our squad has been together for a number of years now and their willingness to listen and learn has seen them develop as footballers,” said Effick.
“They are a talented, hard-working group of boys who understand that they perform even better as a team.
“This was marked by the fact that there were five different goal scorers on Sunday.”
Sebastian Spencer-Perkins opened the scoring for Wimborne who doubled their advantage with an Addwell Chipangura goal shortly afterwards.
Branksome, however, responded in spectacular fashion when Sam Turner fired an unstoppable free-kick over the wall and into the net to reduce the deficit.
But Wimborne hit back with a Jack Ditchburn goal, before Chipangura scored his second of the match.
Wimborne’s dominance in the opening period, was underlined with two further strikes, an own goal and a Mattie Effick goal.
A more evenly-contested second-half saw just one more addition to the scoreline, with Ryan Bailey bringing up the magnificent seven for Wimborne.
Branksome manager Nick Turner was gracious in defeat, while praising his players for reaching the last eight for the first time.
“Wimborne were the stronger team on the day and we wish them luck in the next round,” said Turner.
“This was the first time we have made it to the quarter-final of the cup which is good progress for our team. We passed the ball well at times and created a few goal-scoring chances.”
Wimborne: Ed Phillips, Andrew Cooke, Tommy Rhodes, Ryan Bailey, Sebastian Spencer-Perkins, Jack Ditchburn, Luke Neill, Mattie Effick, Jared Sheppard, Danny Southin, Addwell Chipangura.
Branksome: Charlie Clements, Liam Coogan, Joe Dowty, Ryan Sadler, Jordan Thornton-Dibben, Jamie Wildish, Samuel Turner, Callum Rayner, John Ahern, Miles Nash, Gabriel Kerwick, Ryan Nottle.