Barcelona looked to be heading for a victory after Luis Suarez and Sandro cancelled out Eden Hazard's sublime opener during the second half.
But Gary Cahill headed in from close range in the closing stages - and picked up a bloody nose in the process - to force the game to a penalties.
Alen Halilovic chipped his spot kick against the bar and Courtois saved from Pique as Loic Remy scored the winning penalty.
Chelsea were virtually at full strength, with Hazard, Courtois, Diego Costa, Cesc Fabregas, Oscar, Nemanja Matic, Branislav Ivanovic, Cahill and Cesar Azpilicueta all starting.
Suarez, Ivan Rakitic, Sergio Busquets and Jeremy Mathieu were included in Barcelona's starting XI, while Gerard Pique, Andres Iniesta, Jordi Alba and Pedro were among the substitutes.
Chelsea stormed out of the blocks on the front foot, and centre-back Kurt Zouma almost headed the Premier League champions into a third-minute lead.
Mourinho's men got the goal they deserved seven minutes later thanks to Hazard, who skipped past four Barca players before he coolly slotted the ball underneath Marc-Andre ter Stegen.
The match ebbed and flowed as Barca slowly found their rhythm, but only the woodwork prevented Chelsea from making it 2-0 as Barca's back four struggled to contain the likes of Hazard and Costa.
Oscar's free-kick struck the crossbar in the 26th minute, with Ter Stegen rooted to his line.
Suarez - lively throughout the first half - thought he had equalised in the 31st minute but the Uruguay international was denied again by Courtois, who dived down quickly to his right.
Chelsea were a constant threat on the break and Costa wasted a golden opportunity to double the lead moments later after beating the offside trap and out-running his opponents, but the Brazilian sliced his effort just wide of the post.
Suarez engineered an opening for Rakitic to restore parity before the interval, however the Croatian midfielder skied his shot over the bar from just outside the six-yard box.
However, Suarez got the goal he deserved six minutes into the second half, scooping the ball over Courtois, despite Zouma's best efforts on the line.
Barca dominated from that point and went in front courtesy of Sandro in the 66th minute. The La Masia graduate drifted onto his right foot and curled the ball beyond Courtois.
With time running out in Landover, Cahill courageously headed over substitute keeper Jordi Matip in the 86th minute, picking up a bloody nose in the process.
The goal forced the game to penalties, and after Halilovic hit the bar, Courtois saved from Pique, leaving Remy to calmly seal the win for Jose Mourinho's side.