With Mulligan calling the shots from central midfield, Urlovic doubled Central's lead with a deft flick over an advancing Schischka and then added a second just before halftime when he got his head on Nick Dale's cross.
Mulligan completed Central's scoring right on halftime when he poked the ball into the net from close range.
Tauranga were not bereft of chances, with Jordan Culpepper capping an industrious game when his first-time volley was saved, and Van Der Salm, Josh Nelson and Connor Irvine also going close as City enjoyed a strong 15 minutes midway through the half.
Jindrich Hahn hit the post in the second half and Nelson also forced a good save as Central, who emptied their bench soon after the break, seemed content to head back to Auckland with their squad in one piece.
Van Der Salm said they showed enough yesterday to provide a platform from which to launch a concerted effort for promotion out of the second division over the next month.
"In the league if we concentrated like we did today, we'd be beating teams five or six nil. A cup run is always nice and gives the boys something different but our focus now needs to be on kicking on and taking maximum points from at least the next five weeks to have a realistic shot.
"We play all four teams above us in the next five weeks, so potentially it's season-turning.
"We'll either be a mid-table team with nothing to play for in the last few weeks or give ourselves a shot by picking up wins and putting ourselves up among the top handful."
Van Der Salm admitted City were always going to be up against it against their high-flying opponents, who are contending for the Northern Premier League crown.
"It was hard going down 2-0 early but if we'd taken our half chances maybe it could have brought us back in. With the class they have up front, goals are inevitable and even down 4-0, the boys were still upbeat.
"A result at that stage was probably beyond us but the mindset we played with in the second half bodes well for next week."