The 20-time English champions led 2-0 and 3-1 in Istanbul but could not keep the door shut down the other end, and they remain both of Group A, picking up just four points from five matches.
Galatasaray are third on five points, while Copenhagen are second on five points and in charge of the second qualification spot behind section winners Bayern Munich, with the top two playing out a goalless draw in Germany on Wednesday.
Man United must beat Bayern and hope that Copenhagen and Galatasaray play out a draw in order to advance to the last-16 stage of the competition, but a win would guarantee that they claim a position in the Europa League, so it is set to be a fascinating conclusion to the section.
In Group B, Arsenal secured first position courtesy of a 6-0 win over Lens, while PSV Eindhoven have also advanced to the round of 16, with the Dutch outfit beating Sevilla 3-2 at Estadio Ramon.
Real Madrid claimed top spot in Group C courtesy of their 4-2 success over Napoli, and the Italian outfit will join the La Liga giants in the next round if they win, draw or suffer a one-goal defeat at home to third-placed Braga in their final match in the section.
In Group D, Real Sociedad and Inter Milan have both qualified for the round of 16, and the pair will go head-to-head for top spot in the section on December 12.
Real Sociedad drew 0-0 with Red Bull Salzburg on Wednesday evening, while Inter were involved in a pulsating 3-3 draw with Benfica, and both qualified teams are on 11 points with one game left.
Atletico Madrid and Lazio are through from Group E, with the former at the summit on 11 points, one point ahead of the second-placed Italian outfit.
Diego Simeone's side recorded a 3-1 victory over Feyenoord on Tuesday evening, while Lazio beat Celtic 2-0 to secure their position in the last-16 stage of the competition.
In Group F, Borussia Dortmund are through to the next round following their 3-1 success over AC Milan on Tuesday, while Paris Saint-Germain are second on seven points, with the French champions scoring a late equaliser against Newcastle United in Paris.
Manchester City have won Group G, with the reigning champions claiming 15 points from an available 15, beating RB Leipzig 3-2 on Tuesday, and the German outfit are also through in second place.
Young Boys will drop into the Europa League, with the Swiss outfit beating Red Star Belgrade 2-0 on Tuesday to secure third position in the section ahead of the final matchday.
In Group H, Barcelona secured their spot in the knockout round with a 2-1 success over Porto on Tuesday, while Shakhtar Donetsk recorded a 1-0 victory over Royal Antwerp on the same night.
Porto are second, level on nine points with Shakhtar, meaning that the second qualification position from the section will go to the final matchday, with Porto hosting Shakhtar in what is a gigantic fixture.
Here, Sports Mole selects its standout XI from the fifth game week of this season's Champions League group stages.
DEFENCE
Dmytro Rizynk gets the nod between the sticks due to his excellent performance in Shakhtar's 1-0 success over Antwerp on Tuesday night. The 24-year-old made six saves and three claims during a strong display, with the goalkeeper playing a key role in his side moving onto nine points in their group.
A back four includes Lewin Blum and Joao Cancelo in the full-back positions; the former starred in Young Boys' 2-0 win over Red Star Belgrade, scoring his side's second of the match, while he also made five tackles, contributing to the Swiss team keeping a clean sheet.
Cancelo, meanwhile, was comfortably Barcelona's best player against Porto, finding the back of the net in the 32nd minute before setting up Joao Felix for 2-1, and the Portugal international also had a pass success rate of 90%, completed six dribbles and won four tackles in a truly outstanding performance.
Mario Hermoso and Takehiro Tomiyasu are also included at the back; the former was on the scoresheet in Atletico's 3-1 win over Feyenoord, while he won three aerial duels and two tackles in a strong display.
Tomiyasu, meanwhile, was excellent in Arsenal's 6-0 win over Lens, providing two assists from the right-back spot, while he also boasted a pass success rate of 82% during a strong display in North London.
MIDFIELD
Hakim Ziyech scored twice and provided an assist in Galatasaray's 3-3 draw with Man United on Wednesday evening, with the attacker demonstrating his quality throughout the match, and he was a key contributor to his side claiming a share of the spoils in Istanbul, so there had to be a spot for the Moroccan.
Jude Bellingham also sails into the side once again, moving onto 15 goals for the campaign with a header in Real Madrid's 4-2 win over Napoli, and he then set up Joselu for the home side's fourth late on; it was another outstanding midfield performance from the England international.
There also simply had to be a position in the side for Joao Mario, who netted a hat-trick in Benfica's 3-3 draw with Inter on Wednesday. The 30-year-old scored three times in the opening 34 minutes, while he also completed 47 of his 49 passes in the six-goal thriller in Portugal.
The midfield is completed by Phil Foden, who starred in Man City's 3-2 success over Leipzig, scoring once and providing an assist at the Etihad Stadium, while the England international also used the ball intelligently throughout the match, finishing with a pass success rate of 90%.
ATTACK
Ciro Immobile is unfortunate to miss out due to his brace in Lazio's 2-0 win over Celtic, with Arsenal's Gabriel Jesus and Leipzig's Lois Openda instead featuring in the final third of the field.
Jesus was excellent in Arsenal's 6-0 win over Lens, scoring once and providing an assist, while he completed three dribbles and won three aerial duels during an excellent all-round performance, with the Brazil international showing signs of his best form.
Openda was unfortunate to end up on the losing side during this set of fixtures, as he scored twice in Leipzig's 3-2 defeat to Man City. Not too many players net braces against the Citizens, but the 23-year-old registered two goals in the first half, and he proved to be a huge threat before being withdrawn.
SPORTS MOLE'S CHAMPIONS LEAGUE TEAM OF THE WEEK
Sports Mole's Champions League Team of the Week (4-4-2): Riznyk; Blum, Hermoso, Tomiyasu, Cancelo; Ziyech, Bellingham, Mario, Foden; Jesus, Openda