Diafra Sakho gave the hosts an early lead in east London but Saido Berahino gained West Brom a deserved point with a goal before half-time in front of his new head coach Tony Pulis.
Allardyce's men improved after the interval but were unable to return to winning ways, having ended 2014 with defeats to Chelsea and Arsenal.
However, following a hectic festive schedule, Allardyce believes the run of games in quick succession hampered both teams.
"When fatigue kicks in, more mistakes happen and that happened to both sides," he said.
"The crowd got frustrated but they have to accept that these boys have given everything they've got over the last three games and this one was always going to be one where the energy levels would not be as high.
"That's three big games over Christmas and our points tally was very disappointing in the end, but we have to accept that and try and bounce back.
"We've stopped the losing run with a draw and now we want to go to Everton [in the FA Cup on Tuesday] and get a winning run going again."
Sakho was brought off 18 minutes from time for Enner Valencia, having appeared to hurt his back during a goalmouth scramble.
"We have to have a scan on it and see what condition his back is in," Allardyce revealed.
"Obviously it's a similar situation to when he last came back from [international duty with] Senegal, so on Friday the lads will send him for a scan and we'll see how it goes from there."
