The Bundesliga title-aspirant spurned chance after chance, including three big opportunities in the opening 90 seconds, before Werner's penalty - his 26th goal of the season - early in the second half brought Leipzig a 1-0 victory.
Hamstrung by the absence of the injured Harry Kane and Son Heung-min, Tottenham had little bite in attack, with Lucas Moura ineffective in the central role, but it has a gem in Giovani Lo Celso who twice went close to an equaliser that would have flattered the host.
Tottenham's hopes of another run to the Champions League final look bleak, but Jose Mourinho must know his team could have been effectively out of this tie already ahead of the 10 March second leg in Leipzig.
Patrick Schick shot a yard wide, Angelino rattled the near post from a tight angle on the left, and Werner should have stuck away a close-range chance inside the first minute and a half of a blistering start from Leipzig.
At the other end, Peter Gulacsi had to stretch to turn away a curling shot from Steven Bergwijn that was heading for the bottom right corner.
Schick then headed no more than a foot wide from a corner as Tottenham, with its defence a nervy rabble, again came under siege. Memories came flooding back of its last encounter with German opposition at the Tottenham Hotspur Stadium, a 7-2 unpicking by Bayern Munich in October.
Werner had a glorious chance in the 35th minute after being slipped in to the left of goal, but the Germany international prodded straight at Lloris.
Leipzig finally had an opportunity it surely would not spurn in the 56th minute when Ben Davies hacked down Konrad Laimer, and this time the prolific Werner beat Lloris, his skidding penalty finding the bottom-left corner.
Tottenham almost found a swift equaliser, Gulasci clutching well to keep out Lo Celso's fizzing strike, before Schick missed an even better chance to make it 2-0 to Leipzig.
Gulasci made another fine stop to keep out a free-kick from Lo Celso, who was Tottenham's most impressive outfield player by a distance, before Lucas headed over a glaring late chance.
On a night when he became the youngest ever boss to lead a team in a Champions League knockout match, at the age of 32 years and 211 days, Leipzig boss Julian Nagelsmann saw his give themselves a great change of progressing.