At the 87th minute, Second-tier Coventry looked poised for a victory with a 1-0 lead and dominating play. However, in a sudden turn of events, substitute Djed Spence scored an equalizer just a minute later, breaking Coventry's hopes of securing the win.
Johnson, who had come on as a replacement for the injured Wilson Odobert, then fired past Ben Wilson five minutes later to seal the great escape.
The result came as a big relief to Postecoglou, as until the late goals the home side had looked headed for a deserved upset over their Premier League opponents.
“It was a tough game. Credit to Coventry, I thought they were outstanding,” said the Australian of last season’s FA Cup semi-finalists.
“It was a Cup tie and they really took it to us. We had to just hang in there somehow and we did.”
Brandon Thomas-Asante swept the hosts ahead in the 63rd minute, with Tottenham back-up goalkeeper Fraser Forster looking shaky in his first appearance of the campaign.
Spurs had gone through the first half with more possession but without a single shot on goal and left regulars Son Heung-min, James Maddison and Dejan Kulusevski on the bench.
Kulusevski came on in the 74th for Timo Werner and provided the through ball for Spence while Son got the nod in the 62nd with Maddison, as Postecoglou recognised the need for more creativity.
The win came after two losses in the league, including a 1-0 home defeat by north London rivals Arsenal on Sept 15.
“I don’t think there’s an easy or quick fix to where we want to get to and one result doesn’t change that,” added Postecoglou when asked about the significance of the result.
“Tonight, at least in the end there, we showed what we have been missing in the first four games, that little bit of spirit to find our way to win. We’ve just lacked that little bit of an edge and hopefully tonight gives the boys a bit of belief.”
In the night’s other third-round clash, Brighton & Hove Albion beat Premier League rivals Wolverhampton Wanderers 3-2 on the south coast.
Carlos Baleba opened his account with the Seagulls on 14 minutes and Simon Adingra shot low for the second goal in the 31st.
Goncalo Guedes pulled one back for the visitors a minute before the break with a deflected side-footed shot before Ferdi Kadioglu made it 3-1 in the 84th on the rebound.
Tommy Doyle then lashed home off the underside of the bar in stoppage time for a nervous finish.
The remaining third-round ties will be played next week, with holders Liverpool at home to fellow Premier League side West Ham United on Sept 25.