Tottenham needed two late goals from Djed Spence and Brennan Johnson to beat second-tier Coventry 2-1 and avoid a shock League Cup exit on Wednesday.
Spurs were headed to a third consecutive defeat when Brandon Thomas-Asante fired the Championship side into a deserved lead after 63 minutes.
Tottenham boss Ange Postecoglou had responded to criticism after Sunday’s 1-0 defeat to Arsenal in the north London derby by insisting he will continue his record of winning silverware in his second season at multiple clubs.
However, the former Australia boss was met with a backlash from his own supporters, who booed the decision to replace Swedish starlet Lucas Bergvall early in the second half.
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Coventry nearly shocked Manchester United in the FA Cup semi-finals last season and were sensing an upset when Norman Bassette produced a superb cross for Thomas-Asante to slot in.
As Postecoglou was facing up to a second consecutive early exit in this competition, his blushes were spared when Dejan Kulusevski linked up with Spence, who prodded home for his first Spurs goal in the 87th minute.
Tottenham sensed their moment and in the second minute of stoppage time Rodrigo Bentancur played in Johnson, who rolled home the winner.
“It was a tough game. Credit to Coventry, I thought they were outstanding,” Postecoglou told Sky Sports.
“They really took it to us. We had to hang in there somehow and we did. In the end credit to the lads. It was a typical cup tie.”
He continued: “There is no easy or quick fix to where we want to get to and one result doesn’t change that.
“We showed what we have been missing the first four games. We have lacked a little bit of an edge and hopefully tonight gives the boys a little bit of belief.”
Sky Sports pundit Jamie Redknapp continued: “Spurs were so poor and going nowhere fast. Ange will be mightily relieved. But they have to make sure that this is a turning point for them.
“I didn’t like the team he picked. I’m a big fan of Ange but l thought he took liberties tonight.
“He didn’t select a strong enough team but he has got away with it.
“Tottenham were diabolical up until the goals. Coventry were the better team. Ange Postecoglou will be so relieved.
“Tottenham got away with one, they were very, very fortunate.”
Brighton also booked their place in the last 16 with a 3-2 win over Wolves. Carlos Baleba, Simon Adingra and Ferdi Kadioglu were on target for the Seagulls.