The Bavarians, who also beat Werder in the league last week, will now face RB Leipzig in the final on May 25. Bayern had lost last year's final to Eintracht Frankfurt.
Werder scored twice in two minutes to come back from two goals down to level at 2-2 before Lewandowski hit an 80th minute penalty to secure victory for Bayern, who are chasing a record-extending 19th cup victory.
"We knew it would be tough. We had the penalty but we also had the chances. We deserved to go through," said Bayern's Niko Kovac, who can become the first coach to win the German Cup in consecutive years with different teams.
"We were caught napping in those two minutes and suddenly Bremen are level," said Kovac, who won the Cup with Frankfurt last year. "The penalty may have been a bit lucky but overall we fully deserved the win."
Bayern opened their account with the Bundesliga's leading scorer --Lewandowski -- who stabbed in from close range after a looping header from Thomas Mueller had bounced off the post in the 36th minute.
The hosts had their share of chances but it was Bayern who scored again with Mueller in the 63rd.
Werder, however, fought back through Yuya Osako in the 74th minute and Milot Rashica in the 75th.
Lewandowski, however, silenced the home crowd with his spot kick, which was awarded for a foul on Kingsley Coman by Theodor Gebre Selassie that left the Bremen players furious with the decision.
The Poland forward should have had a hat-trick but he hit the post from a metre out in stoppage time.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar and Toby Davis)
