Goals from Freddie Hinds, Bobby Reid and Niclas Eliasson put the visitors through to the third round. Watford finished with 10 men after Jose Holebas was sent off.
Watford knocked out of FA Cup; third-tier side Cultural Leonesa shock Madrid team.
Like Chelsea, the humble Hertfordshire club, whose only season in European football was 35 years ago, completed their third victory from three games on Sunday, when they beat Crystal Palace 2-1 to add to victories over Brighton and Burnley.
Adem Ljajic scored deep into injury-time to rescue a point for second-placed AS Roma who had to come from two goals behind and with a man down to earn a 2-2 draw at home to Sassuolo.
Currently, players from outside the EU must gain work permits and pass a series of qualification tests to get a "governing body endorsement" and work permit.
Irrepressible AS Roma captain Francesco Totti scored twice to haul the Serie A title challengers back from two goals behind against arch-rivals Lazio and earn them a 2-2 draw in a pulsating derby on Sunday.
Romelu Lukaku struck twice as Manchester United returned to winning ways after two successive Premier League losses with a 2-0 victory over Burnley at Turf Moor on Sunday.
Free-scoring Arsenal replaced Chelsea as Premier League leaders as goals either side of halftime by Theo Walcott and Mesut Ozil helped them recover from a slow start to beat Stoke City 3-1 on Saturday.
AS Roma slipped up in their pursuit of Serie A leaders Juventus as they salvaged a 1-1 draw at mid-table Palermo on Saturday thanks to a second-half Mattia Destro equaliser.
Santi Cazorla scored a 94th-minute penalty as sub-par Arsenal beat Southampton on Saturday with Laurent Koscielny's overhead kick having earlier cancelled out Petr Cech's own goal.
A summary of Sunday's Premier League matches.
Robin van Persie's hat-trick lifted the gloom surrounding Manchester United and manager David Moyes as the English side beat Olympiakos Piraeus 3-0 to overturn a 2-0 first-leg deficit and reach the Champions League quarter-finals on Wednesday.
Southampton's rich vein of form in the Premier League continued as they beat Burnley 3-1 with two goals from Charlie Austin on Sunday while another promoted side, Middlesbrough, fared just as poorly, losing 1-0 at home to Watford.
Manchester City qualified for the knockout stages of the UEFA Champions League with goals from Samir Nasri and Pablo Zabaleta sinking Roma 2-0 to eliminate the Italians.
Chelsea kept up their 100 percent start to the English Premier League season by snatching a 2-1 win at a stubborn Newcastle United on Sunday, grabbing victory thanks to a late own goal by DeAndre Yedlin.
Juventus striker Carlos Tevez scored a magnificent first-half goal, and later missed a penalty, as the runaway Serie A leaders won 1-0 at home to mid-table Genoa on Sunday.
Colombia marked their return to the World Cup after a 16-year absence with an attacking display that earned a 3-0 victory over Greece on Saturday, and the early initiative in what looks to be a really tight Group C.
IMAGES from the English Premier League matches played on Saturday
Fiorentina tore apart Italian rivals AS Roma to reach the Europa League quarter-finals with a 3-0 win that left their opponents' season in tatters and under-fire coach Rudi Garcia tottering on the brink of the sack.
Chelsea manager Maurizio Sarri made a winning start to life in the Premier League as his team enjoyed a 3-0 win over Huddersfield Town at the John Smith's Stadium on Saturday.
Images from the EPL matches played on Saturday
Germany's Thomas Mueller struck twice as the nervy world champions kicked off their Euro 2016 qualifying campaign with a 2-1 victory over Scotland on Sunday.
Mauricio Pochettino's Tottenham Hotspur can cement their status as the top dogs in the capital if they end an eight-year wait and defeat Arsenal in the North London derby on Sunday.
Images from the English Premier League matches played on Saturday
Images from the English Premier League matches played on Saturday
Sean Dyche's Burnley side looked to have secured a crucial point in their fight to move clear of the bottom three but Eriksen broke their hearts with a clinical finish for third-placed Spurs in the 91st minute.
Raheem Sterling scored a hat-trick as champions Manchester City made a convincing start to their title defence with a 5-0 win over West Ham United.
IMAGES from Saturday's English Premier League matches.
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