Coleman was playing just his second first-team game back after being sidelined for 10 months having suffering a horror leg break during Ireland's World Cup qualifier with Wales last March.
Former England manager Allardyce, 63, replaced Ronald Koeman late a year ago, signing an 18-month contract.
Kim Jong Un invites South's Moon to Pyongyang: Seoul
Moon has said in the past that he only wanted to meet the North Korean leader if they can make progress on nuclear issues. The visual was important, a USA official said.
The second-half was a different story as it took less than a minute for Gylfi Sigurdsson to put the Toffees ahead after poor defending, before Oumar Niasse capitalised on more defensive lapses on 51 minutes to double the lead.
Palace pulled one goal back through Luka Milivojevic from the penalty spot in the final stages.
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Teheran's foreign ministry on Thursday summoned South Korea's ambassador to Iran, demanding Seoul and Samsung deliver an apology. Iran had lodged "a strong protest" over Samsung's "dishonest conduct that is contrary to the spirit of the Olympic Games".
"Lots of players haven't played up to their potential that we know they've got this season but we're starting to get better at home".
"There have been a few recent results like getting the draw against West Brom and then beating Leicester where we have seen improved performances but I have come here to push Everton on".
Everton leapfrogged Bournemouth to climb to ninth on 34 points from 27 games and are within two points of out-of-form Burnley in seventh.
Netanyahu speaks to Russia's Putin about Iran
If Israel became more militarily involved in the conflict, what would it mean for the Syrian war - and the wider region? The military says it is "ready for various scenarios and will continue to act according to situation assessments".
"We have to look at our home record and that is where the mainstay of our season lies", Everton manager Sam Allardyce told the BBC. It is something we have to try and turn around. We got 3-0 up and the pressure was off.
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