Recent signing Richard Greenfield made his debut on the right wing, while Pete Featherstone made his first appearance since 27th August at the heart of the defence.
The opening 45 minutes saw very few clear cut chances for either side, with Ryan McBride and Jason Taylor going closest for United as the flow of the game was thwarted by the referee's whistle.
Chris Smith had South Park's best chances as half time approached as he headed over the bar at the far post and forced Grant Young to react well to save with his feet moments later.
Ten minutes into the second half, South Park took the lead as Young produced a fantastic reaction save from a Smith header than ran into the path of Jack Higgins who fired home.
The goal seemed to inspire United as the midfield of Greenfield, Franklin, Tate, MacLean and Taylor took control and began to pick gaps in the home defence.
Tates' cross fell kindly to Taylor who saw his half volley well saved by Wastell in the South Park goal.
United got there equaliser on 69 minutes after a good spell of pressure on the home defence.
Tate produced an inch perfect corner kick to the back post as an unmarked Shaun Loft headed in off the crossbar.
With six minutes of regular time remaining, Tate produced an inviting cross into the area that saw McBride put his body on the line as he beat the advancing 'keeper to the ball and saw his header roll over the line.
McBride was left groggy following the clash and left the action soon afterwards, with Ben Dartnall his replacement.
South Park had one final chance to force extra-time as Loft was penalised for a foul on the edge of the area, but from the resulting free-kick, Loft deflected the effort wide of the goal.
The victory means that United are through to the last 32 of the competition for the first time in 19 years

EUAFC: Young; Featherstone, S Dartnall, Loft, Denton, Franklin, Greenfield, Tate, McBride (B Dartnall 90), Maclean, Taylor
Subs not used: Divall, Harrison, Hughes, Price