Trailing 3-2 on Sunday, Leverkusen were heading for a fourth successive loss in all competitions before Victor Boniface's pass deflected in off Angelo Stiller for an own-goal in the 88th minute.
Jeremie Frimpong then crossed for Schick to head in the winning goal in the fourth minute of added time.
The win brought back memories of Leverkusen's tendency to turn games round in the last minute when they won the Bundesliga title last season.
It also cut Bayern Munich's lead at the top of the standings to six points after Bayern drew 1-1 at Union Berlin on Saturday.
With star attacking midfielder Florian Wirtz out injured, Leverkusen coach Xabi Alonso has had to shake up his tactics and rely more on physically imposing forwards such as Schick and Boniface.Â
The changed system couldn't prevent elimination from the Champions League by Bayern last week and took time to be effective against Stuttgart.
Ermedin Demirovic gave Stuttgart the lead in the 15th minute with a shot on the rebound, and Stuttgart doubled their advantage shortly after halftime with a goal for striker Nick Woltemade.
Frimpong scored to give Leverkusen hope of a comeback, but Stuttgart made it 3-1 when a shot deflected off Leverkusen goalkeeper Lukas Hradecky and midfielder Granit Xhaka for an own-goal.
Leverkusen defender Piero Hincapie scored at a corner soon afterwards as the comeback gathered steam.
Eintracht Frankfurt stayed on course to qualify for the Champions League with a 3-1 win over relegation-threatened Bochum, but only after both teams had to wait nearly an hour for kickoff.
A dispute between visiting Frankfurt fans and the authorities over banners blocking emergency exits led to a standoff, which was eventually resolved in time for the game to kick off 50 minutes late.
Goalkeeper Kaua Santos was the hero for Frankfurt with a spectacular save to keep out a diving header in added time with the score at 2-1, before Frankfurt striker Michy Batshuayi scored for the second week running to secure the win for his new team.
In the other match, Heidenheim ended a nine-game winless run in the Bundesliga as they held off a late comeback from Holstein Kiel to win 3-1 in a battle between the league's bottom two clubs.
