Chelsea's Pedro scored and made a huge impact on his Premier League debut as the champions got their campaign off the ground with an entertaining 3-2 win at West Bromwich Albion while Manchester City won 2-0 at Everton.
James Morrison atoned for an earlier penalty miss to score twice as West Brom looked to capitalise on the 54th minute dismissal of John Terry - who was substituted for the first time ever under Jose Mourinho in last week's loss at Manchester City - but Chelsea held on for a first win of the season.
City maintained their perfect start to the campaign against previously unbeaten Everton at Goodison Park to move top of the table on nine points. They are ahead of Leicester City and Manchester United, who have seven points after three games.
City defender Aleksandar Kolarov opened the scoring on the hour before substitute Samir Nasri collected a deft pass from Yaya Toure to clip a sublime finish past goalkeeper Tim Howard in the 88th.
Chelsea started nervously and the Blues were indebted to returning goalkeeper Thibaut Courtois who saved Morrison's early penalty after Nemanja Matic brought down Callum McManaman.
Sparked by that early scare, Chelsea settled into a good passing rhythm with Eden Hazard linking up with Pedro before the Spaniard squirmed a shot inside the post to open the scoring.
Pedro was then the architect of the second with his angled shot turned in by Costa 10 minutes later.
"He's (Pedro) fantastic," Chelsea's Cesc Fabregas told Sky Sports. "I knew straight away he would make an impact on this team because it was something we did not have.
"A pacy player that could go behind defenders, can dribble, can go one-against-one and stretches teams. I knew straight away the mentality of this guy is fantastic and I'm very happy for him."
Morrison pulled a goal back for West Brom when he fired home through a melee of bodies to halt Chelsea's momentum before Azpilicueta netted his first Premier League goal three minutes before half-time to restore Chelsea's two-goal cushion.
After the break Terry was dismissed for bringing down Salomon Rondon.
Without the team's defensive lynchpin the visitors were vulnerable and Morrison guided a header past Courtois five minutes later to set up a tense final 30 minutes.
Thereafter the hosts laid siege to Chelsea's goal but the visitors held firm and even had chances to win it themselves through Costa, Hazard and substitute Radamel Falcao.
Impressive City make it three from threeManchester City arrived at Goodison Park having scored six goals and conceded none in victories against West Brom and Chelsea and were in a rampant mood versus an Everton side who had collected four points from their opening two fixtures.
City made chances at will with the creative maestro David Silva picking holes in the Everton defence and Sergio Aguero testing Howard with a series of rasping shots.
Having weathered the storm, Everton twice went close with striker Romelu Lukaku having a goal disallowed for offside before clipping the crossbar with a curling free-kick.
City were again on the front foot after the break with Silva hitting the post before the visitors eventually made the breakthrough.
Raheem Sterling drove at the Everton defence before playing an intelligent pass to Kolarov who finished neatly at Howard's near post.
Sterling was replaced soon after by French midfielder Nasri who made sure of the victory with a nonchalant chip after he was played in by Toure.
"We've started this season well but we appreciate that starting the season means nothing because it's a long season," City keeper Joe Hart told Sky Sports.
Elsewhere, Southampton, yet to win this season, were held to a goalless draw at promoted Watford, who have now drawn all three matches since returning to the top flight this season.
Liverpool will travel to meet Arsenal in London on Tuesday (AEST).