The striker wheels away in delight after finding the net for the fourth goal of the game in the clash with the Blades.
Kieffer Moore bagged a double as Phil Parkinson's side produced a stunning fightback from 3-1 down to defeat his former club Chris Wilder's men in a breathtaking eight-goal Championship encounter.
The forward's sweetly-struck volley opened the scoring, but the hosts levelled just two minutes later courtesy of a Tyler Bindon unfortunate deflection.
A quickfire double from Bamford and a Callum O'Hare nod appeared to have put the visitors in a comfortable lead at 3-1, until Moore reduced the arrears with a thumping goal of his own.
Ryan Longman's header early in the second half brought the scores level, paving the way for the goalscorer to give his side the lead for the first time with 14 minutes remaining.
Josh Windass' confident penalty in the closing stages wrapped up the points, halting a run of five games without a win for the promoted side.
When these two sides met in the cup competition back in 2023, it ended in a thrilling 3-3 draw.
This latest clash provided another high-octane, dramatic chapter in their growing rivalry.
Backed by high-profile investors, Wrexham's remarkable rise through the divisions has brought them back to the second tier for the first time since the 1980s.
Ahead of this fixture on the following a poor patch of form, the boss had demanded a return of the famous fighting qualities from those past encounters.
The players undoubtedly delivered, overcoming a 3-1 deficit to secure a memorable 5-3 victory that underscores their never-say-die attitude.