Postecoglou said before the match the expanding football schedule is reaching a "dangerous level" and therefore used it as an opportunity to shuffle his pack, making six changes from the side that beat Brentford in the Premier League on Saturday.
He handed 18-year-olds Lucas Bergvall and Archie Gray their second Spurs starts but, following Dragusin's dismissal, midfielder Bergvall was sacrificed as defender Destiny Udogie came on.
Swede Bergvall, who snubbed Barcelona to join Spurs in the summer, looked forlorn on the bench as his opportunity to show Postecoglou why he should get more starts was cut short.
There were plenty of positives to take for Postecoglou, though.
Valuable minutes for his young players, a clean sheet despite being down to 10 men and confidence-boosting goals for summer signing Solanke as well as Johnson, who reportedly deactivated his Instagram account because of criticism received in the wake of his club's derby defeat to Arsenal earlier this month.
But Spurs' all-out attacking style often leaves them vulnerable to the counter-attack.
Qarabag must have wondered what else they needed to do to find the net as the wasteful Juninho had a goal ruled out for offside, Bayramov went close several times and Vicario produced a brilliant save to deny Cafarquliyev's long-range strike.
The sight of Son hobbling off in the second half could also be a concern as Spurs face Manchester United in the Premier League on Sunday, before a trip to Hungary to face Ferencvaros next week.