Shakhtar Donetsk will play Olympique Marseille and Dynamo Kiev face Paris St Germain with the winners lined up to meet in the semis after the draw was made by UEFA on Friday.
No Ukrainian side had ever previously made it to the UEFA Cup last eight.
Werder will be at home first against Italy's last European contenders Udinese in their quarter-final tie.
City, the world's richest club following their takeover by Abu Dhabi United Group last year, needed penalties to beat Denmark's Aalborg in the last round.
England's last remaining representatives will come up against former Tottenham Hotspur boss Martin Jol, who is now in charge of Bundesliga high-flyers Hamburg.
The UEFA Cup, which will be rebranded as the Europa League from next season, will have a new winner this term after four former champions were knocked out this week.
The quarter-final first legs will be played on April 9 with the second legs on April 16.
The semi-finals take place on April 30 and May 7 with the final set for Wednesday, May 20 at Fenerbahce's stadium in Istanbul.
Draw for the quarter-finals and semi-finals of the UEFA Cup conducted in Nyon, Switzerland on Friday:
Quarter-finals
Hamburg SV v Manchester City
Paris St Germain v Dynamo Kiev
Shakhtar Donetsk v Olympique Marseille
Werder Bremen v Udinese
First leg ties to be played April 9
Second legs to be played April 16
Semi-finals
Werder Bremen or Udinese v Hamburg SV or Manchester City
Paris St Germain or Dynamo Kiev v Shakhtar Donetsk or Olympique Marseille
First leg ties to be played April 30
Second leg ties to be played May 7
