Ange Postecoglou lauded the “real spirit and character” that he admits has been missing from the first four games of Tottenham’s season after late goals from substitutes Brennan Johnson and Djed Spence prevented Coventry City recording another remarkable underdog yarn.

The Championship side deservedly took a lead through Brandon Thomas-Asante midway through the second half that Ellis Simms twice came close to doubling before Spurs recovered to sneak through to the fourth round of the Carabao Cup. At least that is one route to a trophy still in Postecoglou’s sights.

Penalties did for Mark Robins’s team when they lost the Championship playoff final to Luton Town 16 months ago and the FA Cup semi-final to Manchester United last season and they looked set for another trial from 12 yards after Spence, sent in by Dejan Kulusevski, stabbed home from close range in the 88th minute.

Related: Carabao Cup: Preston beat Fulham 16-15 in record-breaking penalty shootout

Yet until the introduction of James Maddison, Son Heung-Min and Kulusevski, Spurs had at times been outplayed – and certainly out-fought – by the spirited second-tier side. Then, in stoppage time, Rodrigo Bentancur played Johnson in on goal and the Wales winger, abused online so badly after his efforts in the north London derby defeat on Sunday that he deleted his Instagram account, slipped in the winning goal.

Both teams may have made middling starts to their respective league campaigns, but the pressure was clearly more on Spurs. Neutrals love a manager who shows ambition and shoots from the hip, after Postecoglou’s conviction that he “always wins things in my second year”.

This he may have done in Australia, Japan and Scotland, where he won the title in his second seasons in charge, but the Premier League is a completely different beast. All of which suggests he will need to win a cup, a feat Spurs haven’t managed since 2008, to maintain his successful second-season streak, and this competition offers an easier target than the FA Cup or Europa League.

Still, he gambled by making eight changes from the side that lost 1-0 to Arsenal, and while he argued that the likes of summer signings Archie Gray and Lucas Bergvall, making their first starts for the club, will benefit from the extended minutes they played here, they could not match Coventry’s tempo in the middle hour of this game.

“It was a typical cup game,” Postecoglou conceded. “Coventry had a real energy to their game. We had to hang in there and it was really hard just to stay in the game. But we [did] and showed some real spirit and character, which has probably been missing in the first four games of the season. But the last 10 minutes we got some really good belief going.”

An audible minority of Spurs fans booed their team off at half-time, and there were indignant cries when Bergvall, the Swedish teenager signed for £8.5m from Djurgården, was replaced by Son just after the hour, when Maddison came on for Dominic Solanke, who showed moments of quality but is still seeking his first goal since joining from Bournemouth for £55m. “I don’t think there are any quick fixes,” Postecoglou said, pointing out that with Qarabag visiting when the Europa League campaign starts next week, Spurs will need to rotate.

Carlos Baleba's thunderous 30-yard strike set Brighton on course for a nervy 3-2 win over Wolves in the Carabao Cup third round.

The Cameroon midfielder rifled home his maiden Seagulls goal in the 19th minute at the Amex Stadium. Yankuba Minteh saw his effort disallowed for offside, before Simon Adingra doubled the hosts' lead, cutting inside and firing into the bottom corner.

Wolves improved after Gonçalo Guedes' 44th-minute strike changed the game's momentum, but were made to pay for their profligacy as Ferdi Kadioglu [pictured] struck to restore Albion's two-goal cushion in a game between two much-changed teams.

Tommy Doyle reignited hope of forcing a penalty shootout by firing home off the underside of the bar in added time, but the hosts held on to maintain their unbeaten start under new head coach Fabian Hürzeler. PA Media

Robins rightly identified Maddison as the difference-maker, as he started dictating possession and taking the sting out of Coventry’s high-tempo approach.

The excellent Jack Rudoni had almost scored after 45 seconds, Fraser Forster saving his shot after playing Bergvall a difficult first touch on the edge of his own area. With Spurs vulnerable down their defensive right, Norman Bassette, a Belgian teenager signed from Caen in the summer, was also outstanding and it was from his low cross that Thomas-Asante, a £2.25million capture from West Brom, adjusted his feet nimbly to sidefoot Coventry into the lead.

Ephron Mason-Clark almost slid home at the near post after Simms’ header from Jake Bidwell’s long cross but Coventry paid for not getting that crucial second goal. “We have to be more ruthless,” Robins said. “Life presents you with chances and we seem to not take those chances.

“You look back at the play-off final and the Manchester United [FA Cup semi-final] and we don’t take the chances, and that’s the case again tonight. It’s disappointing we didn’t see the game over the line. If you’re Tottenham Hotspur, you’re probably relieved to get through tonight.”