The Hammers were relegated from the Premier League in the 2010-11 season but bounced straight back up under Sam Allardyce after spending just one season in the Championship.
O'Neil told Sky Sports News: "We maybe had the edge because we have got a lot of players and the manager and staff who have played and worked in the Premier League before.
"We were only down for a season as well so the lads are used to playing in the league so we maybe took to it quicker than they did.
"We also we had a slightly easier run of fixtures at the start that helped us build up some momentum."
West Ham are currently 12th in the Premier League, six points clear of the relegation zone.