The lively Rani Khedira had a chance on the half-hour mark but he blasted his effort high and wide when well placed in stormy conditions .
Khedira came closer to breaking the deadlock when his free kick crashed against the crossbar shortly before the break.
Manager Harry Redknapp made sweeping changes to his side for the second period, with a new-look team taking to the field.
Poulsen spurned a good chance for Leipzig when he headed wide from a corner seven minutes after the restart.
However, the forward made amends moments later when he outstripped Richard Dunne just after the hour mark before slotting home for the opening goal.
Dominik Kaiser had a chance to double the home side's lead just five minutes later with a volley toward goal, but his shot crept just high of the crossbar, dropping onto the roof of the net.
Frankie Sutherland had a chance to pull one back for the Premier League side in the final minutes of the match, but his header lacked power and was comfortably saved by the Leipzig keeper.