Custodio Castro's first-half goal left the two-leg semi-finals on a 2-2 aggregate.
Braga will meet FC Porto in the first all-Portuguese final in Dublin on May 28.
Unheralded Braga was the underdog but added Benfica to earlier triumphs over Liverpool, Arsenal, Celtic and Seville on its unprecedented run.
Castro, a Portugal defender, rose above the defense in the 19th minute to head in a corner.
Braga, who are third in the Portuguese league, 14 points behind second-place Benfica, were the more composed team as they looked to overturn the first-leg deficit of 2-1.