Ozil, Arsenal's record $70 million signing, ended an eight-match scoring drought stretching back into last season, and also set up Danny Welbeck's first goal for the club in a 3-0 victory over Aston Villa.
The playmaker's influential performance came after manager Arsene Wenger said he should not be a "scapegoat" for a trio of domestic draws and Champions League loss.
The emphatic win ended Villa's unbeaten start to the season, a week after the team stunned Liverpool at Anfield.
West Ham inflicted Liverpool's third loss in five rounds, beating last season's runner-up 3-1.
There were late goals in three of Saturday's games. Under-pressure Newcastle manager Alan Pardew saw his side stage a late comeback to draw 2-2 with Hull. Queens Park Rangers twice came from behind to also draw 2-2 with Stoke, and Victor Wanyama's 80th-minute goal gave Southampton a 1-0 win at Swansea that moved the south-coast club to second place.
Villa started brightly against Arsenal, with American owner Randy Lerner at his first home game in almost two years, before collapsing and conceding three goals in four minutes.
Welbeck, who joined from Manchester United this month, teed up Ozil to run through and roll past goalkeeper Brad Guzan in the 32nd minute. Ozil turned provider by crossing from the left flank and Welbeck powered in the ball from close range. Villa's capitulation was capped by defender Aly Cissokho sliding in to turn a cross from Kieran Gibbs into his own net in the 36th.
At St. James' Park, Newcastle fans displayed banners against manager Alan Pardew, with the team without a league win after five matches.
What will encourage Pardew, though, is Papiss Cisse making an instant impact on his return from knee surgery.
When Cisse went on in the 69th, Newcastle was 2-0 down to Hull after conceding from Nikica Jelavic and Mohamed Diame. The Senegal forward, who had been sidelined since April, scored after four minutes on the pitch, and stabbed in the equalizer in the 87th.
"He shouldn't have played today, really," Pardew said. "My medical staff didn't agree that he should be included today — I think he has trained five days."
At Upton Park, Liverpool showed again just how much it is missing Luis Suarez, who was sold to Barcelona in the off-season.
West Ham took less than two minutes, through Winston Reid's header, to breach Liverpool's defence, and Diafro Sakho chipped the ball into the net five minutes later.
Although Raheem Sterling pulled one back for Liverpool in the 26th, Morgan Amalfitano latched onto Stewart Downing's pass to score in the 88th to secure West Ham's first home win of the campaign.
Liverpool took until December last season to lose three matches before finishing second, but matching that league position seems a far tougher proposition.
But Burnley and Sunderland are still searching for their first wins after drawing 0-0. QPR and Stoke have just one win each after their draw at Loftus Road.
Mame Biram Diouf headed Stoke in front from close range in the 11th minute after Peter Crouch held off defender Rio Ferdinand to nod the ball to his strike partner.
After Steven Caulker levelled just before halftime, Crouch restored Stoke's lead six minutes into the second half, but Niko Kranjcar curled a free kick into the net in the 88th to secure the point.Suggest a correction