The Conference South club held their higher-level opponents to a 1-1 draw in the first encounter but a goal from Ollie Palmer settled matters at Thames Road.
The opening exchanges involved little in the way of clear-cut chances but Mansfield's Chris Clements threatened the Concord goal with a well-struck free kick.
Studer was enduring a nervy start and for a second time, he almost gifted Concord the lead when his clearance fell at the feet of Jordan Chiedozie, who fired the ball wide of the target.
After that opening, Mansfield created the better opportunities, with Ryan Tafazolli heading wide, but Concord were holding their own and deservedly reached the break without falling behind to the League Two side.
Mansfield began to stamp their authority on proceedings after the restart with substitute Fergus Bell striking the post and Simon Heslop going close with a volley, before they netted what turned out to be the winner 16 minutes into the second period.
Rob Taylor made inroads down the flank before providing Palmer with the simple task of tapping the ball into the net from three yards out.
Despite being just a goal ahead, Mansfield remained in full control as Palmer and Matt Rhead both went close to a second, but Chiedozie should have brought Rangers level with eight minutes remaining with the goal at his mercy from eight yards out.
That was the last chance of note for the hosts and that enabled Mansfield to move on through to a second-round date away at Cambridge United.