Liverpool now lead City by a point after Mohamed Salah's hat-trick secured an impressive 4-0 win at Bournemouth.
Chelsea are currently in fourth place, 10 points adrift of Premier League leaders Manchester City after 15 games.
Chelsea and Arsenal are now tied on 31 points in fourth, still eight clear of United in eighth.
Eden Hazard's present deal runs till June 2020.
Goals from Pedro and Ruben Loftus-Cheek helped Chelsea beat Fulham 2-0 in the Premier League on Sunday.
Tottenham showed their quality with a 3-1 win against Chelsea last weekend and sit just five points behind leaders Manchester City.
Unai Emery selected a youthful Arsenal to play in the Ukraine and they romped past Vorskla Poltava 3-0.
The 27-year-old signed a new five-year contract last week and was voted Professional Footballers' Association and Football Writers' Association player of the year for 2016-17.
Dele Alli, who scored his sixth goal in five games against Chelsea, praised Tottenham manager Mauricio Pochettino for his approach.
Victory saw Spurs close back to within five points of leaders Manchester City and three points of second-placed Liverpool.
Eden Hazard has said he is in a quandary about whether to extend his deal or seek a "dream" move to Real Madrid.
N'Golo Kante's new deal will keep him at Chelsea until 2023.
Didier Drogba scored a dramatic late equaliser when Chelsea beat Bayern Munich on penalties in the Champions League final six years ago.
Joe Cole, who turned 37 last week, said his career had been "a dream come true" and he now hopes to move into coaching.
It was a frustrating afternoon for Chelsea, who extended their unbeaten start under Sarri to 18 games.
Chelsea made it four wins from four matches in Group L by edging out a spirited BATE Borisov 1-0 in Belarus.
Alvaro Morata has now scored four in his last four Premier League games.