The 19-year-old hurt himself as he was sprinting back after losing the ball, dropping to the turf with what appeared to be a hamstring problem before being replaced by Brennan Johnson.
Odobert was able to walk off the field, albeit gingerly, and his early withdrawal immediately sparked fears about his availability for a busy end to September for the Lilywhites.
The former Burnley attacker was one of two Spurs wingers forced off the pitch on Wednesday evening, as Timo Werner - who was blasted for his poor display - also failed to last the full 90.
Speaking to the media in his post-game press conference, Postecoglou admitted that he was in the dark about the severity of Werner's issue, but he conceded that Odobert's problem appeared serious.
Postecoglou laments "unfortunate" Tottenham injuries
"Hard to say, Wilson looked fairly significant, so we'll just wait and see," football.london quotes Postecoglou as saying. "Timo, again not really sure if it was just fatigue but yeah, a bit unfortunate for both of those. Obviously, both play in a similar position as well. We're hoping it's nothing too significant."
The Tottenham manager also withdrew regular left-back Destiny Udogie for Djed Spence at half time, but he allayed fears about the Italian, whose early change was pre-planned.
"Yeah, it was always the plan for him to play 45 tonight. He missed a lot of pre-season so it is a balancing act with him to get us some game time because he is not 100 per cent fit. Tonight it was always the plan to play him 45 and play Djed 45."
Odobert was making his third start of the season as part of eight Tottenham changes versus Coventry, whom they dominated possession against without showcasing a killer touch in the final third.
Spurs' toothless attacking play would seemingly prove fatal when Brandon Thomas-Asante opened the scoring just past the hour mark, but substitutes Spence and Johnson spared the visitors' embarrassment in the dying embers.
Postecoglou: 'Tottenham did not dominate, but who cares'
Despite having 70% possession on the night, Spurs registered six fewer shots and only found the target with four of their efforts, leading Postecoglou to admit that his side had to cling on throughout the contest.
Asked if Spurs would have been unlucky to go out, Postecoglou replied: "No, I don't think I've said that so far. If you need further clarification, I said we hung in there so that suggests we dug deep to stay in the game and didn't dominate it.
"I don't fall into the lucky or unlucky because if we've been lucky today, it probably means we've been unlucky in the other games and you know what? Who cares, it doesn't matter. It's a game of football and we found a way to win tonight."
Tottenham now have two full days to recover before hosting Brentford in their next Premier League contest, seeking to improve on a measly record of just one win from their first four games in 2024-25.
Spurs then commence their Europa League campaign at home to Azerbaijani giants Qarabag next Thursday, three days before a daunting trip to Old Trafford to tackle Manchester United.