Antonio Conte was publicly glowing in the aftermath, though the hoarseness of his voice betrayed an evening which had proved rather trying. Perhaps squeezing into a semi-final against Arsenal from such an anxious contest may end up being to his advantage. The cavalry may have come to the rescue in stoppage time, but at least the head coach can point to his much-changed side’s spluttering display as further evidence the squad lack the required depth to compete on all fronts. The board may still have to dig a little deeper when it comes to next month’s transfer window.

This proved to be a slog against a spritely Bournemouth team, whose own display improved markedly as the night progressed, and Chelsea’s initial composure degenerated into something more ramshackle. All the early dominance drained away, for all that Ethan Ampadu excelled alongside experienced team-mates at the heart of the three-man defence. The visiting side merited reward and thought they had claimed the extra half-hour when in the confused aftermath of a long throw, Jordon Ibe calmly squared the ball across the penalty area and Dan Gosling, granted too much space by a dithering Davide Zappacosta, whipped a fine first-time shot beyond Willy Caballero.

That had been reward for a one-sided second period when the tightness of the tie had eventually forced Conte to turn to his elite performers. Ultimately they performed to the last, Bournemouth appearing to switch off in anticipation of extra time to permit Chelsea to counter at pace.

There was a sumptuous backheel from Eden Hazard to flummox Simon Francis, and a finish from Álvaro Morata which squirted under Artur Boruc and into the net. Conte should have been delighted at that, though he missed his first-choice striker being booked for his celebration in the din that followed. It was Morata’s fifth caution of the season and will see him suspended for the trip to Goodison Park on Saturday. “I must be honest, I didn’t see this yellow card,” said Conte as he digested the news post-match. “He’ll be a big loss for us, especially because we have to play a tough game against Everton. It’s a pity, but we have to accept it. Tonight, many players were booked.”

It was a spiky occasion and one which, for all the relief of victory, will have left the head coach privately frustrated. His fringe players had been granted an opportunity but, in truth, the vast majority struggled to take their chance.

Michy Batshuayi’s display was peripheral and after recent ankle problems, rusty in front of goal. Danny Drinkwater’s campaign has been even more disrupted by injury, so perhaps his rather tentative display was to be expected, but there was also the sight of Willian and Pedro fading having been involved in three games in a week. The Brazilian had been bright in the opening period, converting his side’s opener crisply into a gaping net after the impressive Kenedy’s own backheel and Cesc Fàbregas’s square pass. But his sixth club goal of the season was the prelude to a sloppy performance that had Conte fidgeting in frustration on the edge of his technical area.

The Italian has suggested he is yet to speak with Marina Granovskaia and the club’s recruitment department about plans for next month but having maintained Chelsea’s interest on all four fronts into the new year, he will expect significant additions to be made.

At least he will be able to rely more on Ampadu, with the 17-year-old excelling having made his only other start for the club in central midfield. The Wales international’s concentration was maintained despite a booking inside 90 seconds, his challenge on Jermain Defoe potentially damaging the striker’s ankle ligaments to hamper Bournemouth’s Premier League survival bid yet further. The forward hobbled on until the quarter-hour mark before succumbing to the pain.

Yet Ampadu’s display merely improved, “Despite having to play 93 minutes with this danger over him,” said Conte, who purred at one superb block to choke Lys Mousset’s attempt. “He showed great maturity and personality. Now he has to stay calm because maybe, in the next game, he’ll go back to playing for the youth team. I don’t know. But this player is very humble. He could have a great future.” He will surely feature in the semi-final, with Bournemouth’s impressive rally after the interval – driven by Ibe’s energy – having effectively been staved off. Their equaliser proved a deception, with the manager Eddie Howe left to bemoan the game’s “heartbreaking” finale.