Younousse Sankhare's first goal since October put Bordeaux in front in Wednesday's semi-final tie.
But Strasbourg produced a superb second-half comeback, with Ludovic Ajorque's fifth goal in as many games preceding a match-winning double from Mothiba.
Jimmy Briand got a late goal back for Bordeaux, but it is Strasbourg who will face Guingamp in the final, which takes place in Lille on March 30.
Paris Saint-Germain have won the competition for the last five years, but they lost to Guingamp in the last eight.
Progression to a national cup final highlights the strength of Strasbourg's astonishing rise back through the French league system following their liquidation in 2011 and evenutal reinstatement in the fifth tier.
Strasbourg made a bright start with Stefan Mitrovic flicking an effort just over and Mothiba testing Benoit Costil.
But Bordeaux struck first after 14 minutes when a poor punch from home goalkeeper Bingourou Kamara allowed Sankhare to volley in from 10 yards, despite Kenny Lala's attempt to clear off the line.
Ajorque's acrobatic effort from close range failed to find the target for Strasbourg, whose frustration increased when Adrien Thomasson and Ibrahima Sissoko had separate penalty claims denied shortly before half-time.
There was relief for the hosts when they levelled just four minutes into the second half, with Lala's right-wing cross pushed out by Costil as far as Ajorque, whose deflected effort from eight yards beat the keeper.
Strasbourg moved ahead in the 55th minute, Ajorque and Lala combining to set up Mothiba for a first-time finish from six yards, and a rapid turnaround was complete on the hour mark.
Mothiba was the hero again, heading in from inside the six-yard box after Ajorque headed a Lala free-kick back across the face of goal, and a shaken Bordeaux side offered no response until Briand converted substitute Francois Kamano's cross with eight minutes left.
Nuno da Costa hit the crossbar for Strasbourg before they held on for a nervous final few minutes, the Stade de la Meinau erupting at the full-time whistle.