The 30-year-old midfielder turned in a rebound after an Everton counter-attack as a toothless Chelsea were stung on a break by Dwight McNeil and Dominic Calvert-Lewin.
Substitute Dobbin struck from the edge of the area as Chelsea failed to clear to seal victory in extra time.
Chelsea's misery was worsened by a first-half injury to captain Reece James, who can't get over his fitness woes. Robert Sanchez also potentially came off injured.
Nizaar Kinsella was in Liverpool witnessing yet another Chelsea setback.
Robert Sanchez - 6
He was shaky coming for crosses but made a top save from for Dwight McNeil before conceding later on.
Reece James - 5
He was substituted after just 26 minutes, having barely made an impact. It's a real shame he might again be injured.
Axel Disasi - 7
Both he and Benoit Badiashile bullied Dominic Calvert-Lewin physically but got caught out with some long balls.
Benoit Badiashile - 6
He put Sanchez under pressure early on but grew into the game and was pretty dominant. He could have been tighter to Calvert-Lewin, who saw his shot deflected before Doucoure turned in the rebound.
Marc Cucurella - 5
He gave away a soft free-kick early on, the exact issue Pochettino warned against. He was also poor in possession, albeit out of position at right back, after James came off injured.
Enzo Fernandez - 5
The Argentina international had a good shot saved early on, playing as a no10; he was tidy but unthreatening and substituted in the 66th minute.
Moises Caicedo - 5
He was not as slick as usual, and some may ask where he was for Everton's transition leading to the goal.
Conor Gallagher - 5
The returning midfielder should have taken a yellow card for the team after allowing Dwight McNeil to bypass him and set up the winning goal.
Cole Palmer - 7
Shot at every opportunity and dictated play well, doing his job as the ball-striker-in-chief.
Armando Broja - 6
After a few understandably rusty touches, Broja began leading the line well in a physical battle with his opponents. Unfortunately, he had little service to work from and was taken off.
Mykahilo Mudryk - 6
The Ukrainian's dribbling ended Ashley Young's night early, forcing him off for Nathan Patterson before half-time. Somehow both a major threat and wasteful.
Levi Colwill - 6
Roamed forward with a good left-back run early in the first half, but maybe could have got back for Everton's goal quicker.
Nicolas Jackson - 5
Fluffed a 78th-minute sitter, kicking thin air instead of the ball. Also had another chance after coming on.
Raheem Sterling - 7
The England international got 30 minutes in front of Gareth Southgate and create the best chance, for Jackson, which was missed.
Djordje Petrovic - 4
Debuting in hugely difficult circumstances, Petrovic's only action was failing to clear a corner and seeing Dobbin slam a great finish in extra time to punish his mistake.
Ian Maatsen - N/A
Unused subs Silva, Gilchrist, Matos, Castledine