Manchester United made a winning start under Jose Mourinho and Liverpool won 4-3 at Arsenal in a seven-goal thriller on Sunday.
Zlatan Ibrahimovic scored on his Premier League debut as United won 3-1 at Bournemouth.
Juan Mata put United ahead after a poor back-pass and Wayne Rooney doubled their lead just before the hour mark.
Ibrahimovic then put the game beyond doubt before Adam Smith scored a fine consolation for Bournemouth.
"It's a good victory and an important victory, at a place where we lost three points last season," United manager Jose Mourinho told Sky Sports.
"It is just three points and there is no time to breathe because the next match is coming soon."
The arrival of Mourinho at United seemed likely to signal the end for Mata, having been sold to United by the Portuguese when at Chelsea.
"I'm really happy with him but I'm really happy with the whole team," United manager Jose Mourinho told Sky Sports.
"We are building something strong and to build something strong, it's not just the people on the pitch, it's the people on the bench, the ones not selected and fundamentally it's the Red family."
Theo Walcott missed a penalty for Arsenal early on after a foul by Alberto Moreno but made amends with the opening goal after 31 minutes.
But Philippe Coutinho equalized with a brilliant 25-yard free kick right on half-time and Liverpool then roared clear in the first 10 minutes of the second.
First Adam Lallana clipped a lovely finish for the second goal then Coutinho finished off Nathaniel Clyne's cross.
Arsenal looked shell-shocked and Sadio Mane made it 4-1 in his first Premier League game for Liverpool with a stunning run before slamming in a superb left-foot finish on 63 minutes.
Alex Oxlade-Chamberlain pulled one back soon after and there were still 15 minutes to go when Callum Chambers headed Arsenal's third.
But Liverpool held on to clinch victory and join United, Manchester City, Swansea and West Brom as victors on the opening weekend.
"We started the game not so brightly," Lallana said. "I thought we came out after half-time brilliantly.
"Arsenal are a top team. If we want to achieve something special this year we have to be more disciplined defensively as a unit.
"I think we maybe showed them a bit too much respect in the first half.
"The manager told us how fit we were and we came out of the traps flying."
Coutinho said Liverpool will take a lot of heart from the win.
"It gives us confidence, for the whole team," he said. "We played a difficult game, played really well and we have to go step by step and keep going."