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.
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.
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.
