The match was played amid a heavy downpour, with Leeds hitting a post through Lubomir Michalik's shot and seeing an effort cleared off the line in the second period as they searched for a leveller.
In Sunday's other game, Notts County sacrificed a great opportunity to reach the third round and a tie against local rivals Eastwood Town when they were held 1-1 by Kettering Town at Meadow Lane.
It cannot, however, be denied that Kettering were full value for earning a reply at Rockingham Road on Tuesday week.
Notts were never on top throughout the game and, the longer it went on, the likelier Kettering seemed to snatch a winner.
In the 18th minute the Poppies took the lead thanks to a dreadful defensive error by County's Jamie Clapham.
He presented the ball to Alfie Potter who picked out Brett Solkhon in the area and the midfielder, who had made a great run, finished off well to beat Russell Hoult.
Notts got on terms in the 34th minute with one of the best bits of football they produced in the entire match. The brains behind the goal was Jamie Forrester who drilled a lovely low ball in for the unmarked Sean Canham to score with ease from six yards.
