But Liam Plunkett (5-52) claimed his maiden five-wicket ODI haul and Adil Rashid took 3-34 as England earned a 2-0 lead with two games remaining.
Then Ali took over, reaching his century off 53 balls, the last of them being his eighth six.
"I enjoyed myself", Ali told Sky Sports. "I know there will be tough times ahead, and I try not to get too over-confident, but I've played quite a bit for England now and the experience has allowed me stay focussed on what's to come".
But, with Ali leading the charge, England scored 123 runs in their last 10 overs after being sent into bat by West Indies captain Jason Holder.
His century also included seven fours, with the left-hander needing a mere 12 balls for his second fifty.
He should have been out for 87 when he smashed a Jerome Taylor full toss to point only for fit again Chris Gayle, back from a hamstring injury, to drop the catch.
Joe Root with 84 and Ben Stokes (73) were the other main contributors to an imposing total of 9-369 off 50 overs.
Ali went to his century with two sixes in three balls off Miguel Cummins - a pull over deep square leg followed by a soaring drive over long-on.
Following a washed-out second ODI at Nottingham's Trent Bridge on Thursday, Stuart Law's men only have three more matches in England - starting with Sunday's fixture in Bristol - and a three-match ODI series in New Zealand scheduled before the qualifiers.
"Most people would be excited to have a guy like Andre Russell (who is due to complete a one-year ban for a missed drugs tests in January) also champing at the bit to play worldwide cricket".
West Indies opener Evin Lewis hit two sixes in David Willey s first over before the left-arm quick had him caught at mid-on by Ali.
"We need to be better in the field". Bristol is not a happy hunting ground for West Indies; they are yet to register a win in three matches.
Any conflict around Kurdish northern Iraq could have ramifications across the Middle East, not least in Syria, Turkey and Iran. Plans to retake the town of Hawija there have been complicated by political wrangling among Iraq's disparate security forces.
Fellow new recruit Paulinho then became the unlikely hero with a goal that issued a considerable statement to his doubters. But Denis Suarez and Paulinho struck to give Ernesto Valverde's side their fifth win on the bounce across all competitions.
New loan signing Sanches, of Bayern Munich , had a Premier League debut to forget in the 1-0 defeat to Newcastle last time out. Dortmund lost the 2013 Champions League final at Wembley, and Tottenham had been a woeful 1-8 in their adopted home stadium.
While at Manchester City, Barry played 132 games and made a further 131 appearances across his two spells at Everton . "No. Paul Ince is backing Arsenal to ease to an easy win over West Brom at The Emirates on Monday night.
It is probably nothing more than a coincidence that Arsenal and Chelsea played in the Premier League recently nearly an exact year from what many people see as the turning point in last season´s race to the EPL title.
Miller gave thanks to Atwood, "who created this world for all of us", then capped the night by saying, "It's been lovely". Waithe also made history when she became the first black woman to win an Emmy for comedy writing.
Traffic came to a standstill as thousands of workers blocked the streets escaping their office buildings. Market stall vendor Edith Lopez, 25, said she was in a taxi a few blocks away when the quake struck.
Former FBI Director Mueller was appointed by the Department of Justice to investigate Russian interference in the USA election. There's more to the $100,000 in ads that Facebook sold to a Russian propaganda company.
Anissa Weier trembled as the jury's verdict was read after a week of testimony and some 11 hours of deliberations. Both Weier and Geyser were originally charged with being a party to attempted first-degree intentional homicide.
Interior chief: Shrink 4 national monuments
Originally designated by President Clinton in 2000, President Obama expanded Cascade-Siskiyou National Monument in January 2017. But the head of the nonprofit Friends of Gold Butte vowed a legal challenge of any boundary change by presidential edict.