Enzo Maresca's side have squandered eight points from winning positions already this season and the latest let-down came at Stamford Bridge when Sunderland's Chemsdine Talbi's snatched a stoppage-time winner.
Chelsea had led through Alejandro Garnacho's fourth-minute strike, his first goal since arriving from Manchester United, but the Black Cats equalised in the 22nd minute through Wilson Isidor.
The Blues were unable to regain the lead despite dominating possession for long periods as they delivered the latest in a growing list of toothless displays.
Chelsea are languishing in mid-table in the Premier League after their four-game winning run in all competitions came to an end.
The west Londoners had hoped to build on the momentum of qualifying for the Champions League and winning the UEFA Conference League last season before lifting the Club World Cup.
Instead, left-back Cucurella conceded they have plenty to work on, starting with finding a more ruthless edge.
"It's a really tough one to take," Cucurella told Chelsea's website. "We need to improve with these matches. I think we need to learn to manage the game better.
"We scored early, dominated at that stage but then we let the opponents back into it. We need to learn to kill games off when we are on top.
"We need to learn to be better against teams that come with a game-plan such as Sunderland where they stay very narrow and compact."
For the second time this season Chelsea were undone by a long throw-in as Nordi Mukiele's launch into the six-yard box created havoc and Isidor poked home the equaliser.
"We know that a lot of teams are trying to use long throws more and we need to manage them better," Cucurella said.
"We worked on them yesterday because we knew Sunderland will look to use long throws and be direct from set-pieces.
"I think maybe we relaxed a little bit and they punished us. It's no excuse. We need to learn from this and make sure this doesn't happen again."
