Bell made 77 while captain Michael Vaughan chipped in with 56 as England, having bowled the hosts out for 195 off 49.3 overs, reached 197-5 off 48.4 overs in the first of their four meetings.
Bell and Vaughan - who said on Saturday he would quit the tour if he had to shake the hand of Zimbabwean President Robert Mugabe - put on a stand of 111 runs for the second wicket after Vikram Solanki went for seven.
However England suffered a minor wobble when they lost Bell and Paul Collingwood for just five runs leaving them at 175-5 but Kevin Pietersen and wicket keeper Geraint Jones guided them to victory against a Zimbabwe side that can be proud of its performance given their inexperience.
Earlier Zimbabwe's innings had been rescued by 18-year-old Elton Chigumbura who shared an 82-run partnership with Dion Ebrahim with the youngster scoring 52 and bring up his third One-day half century.
England's veteran seamer Darren Gough - who admitted he had only come on the tour reluctantly because of the human rights abuses of Mugabe's regime - returned the best figures for the visitors of 3-34.
He had helped kick some life into England's early play after Vaughan decided to field having won the toss.
England were struggling to make progress early on against a Zimbabwe side who have not won a match since the controversy which led to 15 rebels walking out on the team alleging racism when it came to selection policies.
However Gough, then Alex Wharf put England in the driving seat by claiming important breakthroughs to leave Zimbabwe struggling to set a competitive target having slumped to 90-5 after 30 overs.
Zimbabwe's opening stand saw them notch up 36 runs in nine overs but Gough, England's most experienced One-day player, kept that in check when he tempted Stuart Matsikenyeri into driving his slower ball but instead looped the ball gently to Pietersen at cover.
Two overs later Wharf took his first wicket just five balls after being introduced as a replacement for Gough when Taylor, who had progressed to 17 off 31 balls, attempted to cut a short delivery but instead chopped onto his stumps.
Wharf claimed his second wicket in similar fashion, removing Vusumisi Sibanda six overs later when he rocked onto the back foot and attempted to force through the covers but instead gloved the ball onto his stumps.
England's dominance was boosted with the brilliant fielding of Andrew Strauss at backward point to remove stubborn middle-order batsman Mark Vermeulen once Gough had returned to the attack.
Four overs later Gough struck again to dismiss captain Tatenda Taibu, who made just five before he became the third top-order batsman to play-on.
England, who flew into Harare from Johannesburg on Friday, had been reluctant to pull out of the tour despite the refusal of Mugabe's government to grant visas to 13 journalists assigned to cover the event.
An 11th hour U-turn to grant them entry saved the tour, but the decision came too late for Friday's opening match in Harare, which was cancelled, resulting in the One-day tour being cut back from five matches to four.