Bremen were hanging on for the 1-1 draw which would have been enough for them to reach the last eight of the competition.
But with the seconds ticking away, Wiese, who had been outstanding for the Germans with a series of superb saves, spilled a straightforward catch to give Emerson the easiest of tap-ins.
Wiese's howler left the aggregate score at 4-4 and Juve advanced on the away goals rule.
French midfielder Johan Micoud struck for Werder in the 13th minute with a delicate chip, before David Trezeguet levelled the score midway through the second half with a typically clinical finish.
Juve coach Fabio Capello was able to recall strikers Zlatan Ibrahimovic and Trezeguet, both absent from Saturday's 1-0 league win at Sampdoria due to injury.
The return of the club's first-choice attacking partnership, with 32 goals between them in all competitions this season, forced Alessandro Del Piero, Juve's all-time top scorer, onto the substitutes' bench.
The crowd was swelled by an extra 6,000 with Juve granting free admission to volunteers at the Winter Olympics.
Another keen spectator was Italy coach Marcello Lippi, keeping a watchful eye on his World Cup hopefuls.
Juve set the early pace with Pavel Nedved's dangerous cross cut out by a Frank Fahrenhorst's outstretched boot and Fabio Cannavaro heading the resulting corner over the bar.
But the home side's early swagger quickly diminished.
Juve's defence tried to play the offside trap when Christian Schulz chipped into the penalty area, but Micoud timed his run to perfection and clipped the ball right-footed past the advancing Gianluigi Buffon.
A mistake by Schulz almost brought Juve back on level terms in the 23rd minute, when he almost headed into his own net when trying to cushion a header back to Wiese.
Fortunately for Schulz, the ball rolled wide and from the corner Wiese pulled off a miraculous reflex save to keep out Ibrahimovic's close-range volley.
Ivan Klasnic went close to extending Werder's lead two minutes before the break, but Buffon was equal to his rasping drive.
Wiese produced a jaw-dropping double save to deny Nedved and then Emerson as Juve ended the half with a flourish.
Capello brought on Alessandro Del Piero for Ibrahimovic and Adrian Mutu for Mauro Camoranesi and the switch added impetus to a previously sterile attack.
Juve kept plugging away and they equalised in the 65th minute.
A robust run by Nedved caught Werder napping and as the ball started to run a little too far ahead of the Czech midfielder, Trezeguet came steaming in to fire home.
Wiese palmed away a Mutu free-kick and an Emerson piledriver, but his luck cruelly deserted him two minutes from time with an error that is likely to haunt him for the rest of his career.
: Buffon, Zebina, Thuram, Cannavaro, Zambrotta (Balzaretti 70), Camoranesi (Mutu 57), Emerson, Vieira, Nedved, Ibrahimovic (Del Piero 57), Trezeguet.
: Wiese, Owomoyela, Naldo, Fahrenhorst, Schulz, Frings, Baumann (Pasanen 73), Micoud, Borowski, Klose, Klasnic (Haedo Valdez 81).
: G Poll (England).