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Arsenal claimed a first win in six matches in all competitions with defender Gabriel scoring the only goal in 1-0 victory at Wolverhampton Wanderers.
Klopp's team posted 3-1 win over Crystal Palace.
The Hammers moved within a point of fourth-placed Arsenal.
City extended their lead over Chelsea, who take on third-placed Liverpool on Sunday, to 11 points, while Arsenal are fourth on 35 points.
Manchester United suffered a fourth defeat in five Premier League games as they went down 4-1 at Watford to pile more pressure on manager Ole Gunnar Solskjaer.
A second-half goal from Emile Smith-Rowe gave them a 1-0 home win over Watford.
Goals by Emile Smith Rowe, Pierre-Emerick Aubameyang and Bukayo Saka crowned a majestic first-half display by Arsenal who were sharper and hungrier than a listless Tottenham side.
The Gunners have lost their opening three league games of a season for the first time since 1954-55.
Lukaku needed only 15 minutes to get off the mark for the Blues after his reported 97.5 million pounds transfer from Inter Milan.
The Belgium international is in line to feature against Arsenal in the Premier League on Sunday.
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