Seconds after Bournemouth had a chance to snatch victory the ball went back down the other end and Sterling's deflected shot looped past helpless Bournemouth keeper Asmir Begovic.
Christian Benteke condemned struggling Sunderland to a 1-0 defeat and sent Liverpool level on points with sixth-placed Manchester United in the last Premier League game of the year on Wednesday.
Burnley's 0-0 draw at home to Tottenham Hotspur in the Premier League on Sunday completed a profitable weekend for the three clubs bidding to avoid a swift return to the second tier of English football.
Zlatan Ibrahimovic was again at his irrepressible best with another double earning Jose Mourinho's increasingly confident Manchester United a 2-0 win at West Bromwich Albion in the Premier League on Saturday.
Leicester City's Premier League title defence got off to a terrible start as they were beaten 2-1 at promoted Hull City, who shrugged off an injury crisis to shock the champions on the opening day of the new season on Saturday
Manchester United survived a tense finale to beat West Bromwich Albion 2-1 at the Hawthorns on Sunday and close to within 11 points of Premier League leaders Manchester City.
Manchester City stayed ahead of age-old local rivals Manchester United at the top of the Premier League on Saturday with a deserved 1-0 win away to champions Chelsea.
London clubs' erratic start to the Premier League season improved with victories for Chelsea, Tottenham Hotspur and Arsenal on Saturday but Manchester sides continued to make the running, with City and United the top two.
Chelsea moved 12 points clear of Arsenal, who suffered their fourth defeat in nine league matches, while fourth-placed Liverpool fell 13 point adrift after their wretched 2017 continued with a shock 2-0 defeat at resurgent Hull City.
Fiorentina fell behind to a spectacular scissors-kick goal from Pandurii's Eric Pereira before hitting back in the last five minutes to win 2-1 on Thursday and lead a charge of teams into the Europa League knockout stages.
A stoppage-time goal from Martin de Roon earned Middlesbrough an unlikely 1-1 draw at Premier League leaders Manchester City on Saturday.
Aston Villa were relegated from the Premier League for the first time since it began in 1992 when they suffered their 24th loss of the season in a 1-0 defeat at Old Trafford on Saturday.
England striker Daniel Sturridge celebrated his return after a five-month injury layoff when he came off the bench to score as Liverpool beat West Ham United 2-0 in the Premier League on Saturday.
Sergio Aguero scored five times in 20 minutes including an eight-minute hat-trick as Manchester City came from behind to thrash Newcastle United 6-1 and go back to the top of the Premier League table on Saturday.
Manchester United teenager Marcus Rashford wrote another chapter in his fairytale introduction to top-flight football, scoring a brilliant goal to secure a 1-0 win in his first derby match against Manchester City on Sunday.
Cesc Fabregas and Branislav Ivanovic struck late goals to earn improving champions Chelsea a 2-1 victory at Southampton in the Premier League on Saturday.
Leaders Chelsea stretched their Premier League winning run to a club record-equalling 11 as Diego Costa's header shortly before halftime secured a 1-0 victory at London rivals Crystal Palace on Saturday.
Manchester United won 2-1 at Crystal Palace to consolidate fourth spot.
Manchester United maintained their push for a Champions League place after a Juan Mata goal on Saturday pushed Norwich City to the brink of relegation.
Images from the English Premier League matches played on Saturday
Second-half goals from Willian and Diego Costa helped Chelsea ease past Hull City 2-0 on Saturday and grab their first win in four Premier League games.
Chelsea were given a stiff examination by Stoke City at Stamford Bridge on Saturday before extending their lead at the top of the Premier League.
In England, the celebrations went on at Leicester City as they marked their fairytale Premier League success with another win while champions Juventus' long unbeaten streak in Italy came to an end.
Cristiano Ronaldo bagged his first five-goal haul in Real Madrid's 9-1 hammering of Granada while title-chasing Bayern Munich and Chelsea won again in Germany and England.
'Keep Calm And Carry On' could be Claudio Ranieri's motto for the second half of the season after Leicester City earned a 1-0 win at Tottenham Hotspur on Wednesday to put the heat back on Premier League leaders Arsenal.
'We know many things are going to happen.' 'People should be preparing for sea level rise, for increased cyclonic activity, for drought.' 'One reason I wrote the book is to alert people to the dangers that they face.' 'For example, Mumbai faces enormous threat.'