Sancho slotted home the winner after the visitors launched a counter-attack from their own area in the second half of the quarter-final at Borussia Park.
Leipzig fought back with second-half goals by Christopher Nkunku and Yussuf Poulsen before Sorloth's winning header in the 93rd minute after Gladbach had led 2-0 at half-time.
Champions League: Pep Guardiola demands more as Manchester City cruise past Borussia Moenchengladbach
Premier League leaders City camped inside the Gladbach half for most of the game, which was moved to Hungary due to Germany's COVID-19 travel restrictions.
Champions League: Coach Marco Rose urges Borussia Moenchengladbach to be brave against Manchester City
City's 1-0 win over Arsenal on Sunday extended their record run in all competitions, while Gladbach are struggling in the Bundesliga after taking a point from their last three games.
Bundesliga: Marco Rose vows not to take Borussia Moenchengladbach stars with him to Borussia Dortmund
"If Marco Rose says he will not take a player with him to Dortmund, then he won't take a player to Dortmund. Done and dusted," said new Borussia Dortmund coach Marco Rose.
A long-term target of Dortmund's, Rose has also been linked to the Manchester United job in the past.
Liverpool's Champions League last-16 away tie against RB Leipzig will now be held at Puskas Arena in Budapest. The change of venue happened because coronavirus restrictions bar Liverpool from travelling to Germany.
Manchester City's trip to Borussia Moenchengladbach is scheduled for 24 February but the GErman club are studying their options.
Champions League: Liverpool's last-16 away game against RB Leipzig to be moved to Budapest, say reports
RB Leipzig are in discussions to host the game at the Puskas Arena in the Hungarian capital, among other options.
Bundesliga: Bayern Munich stay seven points clear at top after win over Hoffenheim; Borussia Dortmund also claim victory
The European champions exacted perfect revenge for their only defeat of 2020 after losing by the same 4-1 scoreline at Hoffenheim last September.
The victory was Leverkusen's first in five league games and puts them second on goal difference, level on 32 points with RB Leipzig, who host Union Berlin on Wednesday.
For the second consecutive Saturday, Leipzig, who lost 3-1 last weekend at home to Borussia Dortmund, missed out on first place.
What will you think of when you think of 2020? How will you remember sport in a year defined by the pandemic? Here are 20 moments — captured in images and videos — that encapsulate sport in a year like no other:
Bundesliga: Robert Lewandowski's brace takes Bayern Munich to top of table; Marcus Thuram sent off after spitting on opponent's face
Lewandowski, who beat Cristiano Ronaldo and Lionel Messi to the individual award on Thursday, scored in the 93rd minute to put the reigning champions two points clear heading into the winter break.
Champions League: Real Madrid and Zinedine Zidane brace for crucial tie versus Borussia Moenchengladbach
Should Real Madrid suffer a humiliating group stage exit, ending their remarkable run as the only team to ever avoid elimination at this phase, coach Zinedine Zidane's future will be at risk.
UEFA Champions League reaches a crucial final matchday in the group stage with fates of Real Madrid, Inter Milan, Atletico Madrid and Manchester United still unclear.
As per measures announced on Wednesday, all professional sport, including the Bundesliga, must be played behind closed doors until at least the end of November.
Dortmund had three times as many shots on goal and dominated with 80 percent possession, which left their Swiss coach Lucian Favre to admit, "we are very disappointed, it's hard to accept this defeat".
Bundesliga: Erling Haaland double inspires Borussia Dortmund to win over Monchengladbach; Hertha Berlin thump Werder Bremen
Coach Lucien Favre banked on youth by starting two 17-year-olds in Reyna and England Under-21 midfielder Jude Bellingham, who made his league debut, with Sancho and Haaland, both 20, leading Dortmund's attack.
Bundesliga: RB Leipzig reclaims third place after win at Cologne, in line to seal Champions League spot
Leipzig are now third on 58 points, two behind second-placed Borussia Dortmund and nine off leaders Bayern Munich