The Hammers impressed in their opening-day victory over Cardiff, leading to predictions of a top-half finish, but their manager insists there is no room for complacency
Allardyce's men were 2-0 victors over the Premier League newcomers on Saturday thanks to goals from Joe Cole and Kevin Nolan.
|66/1||Andy Carroll is 66/1 with William Hill to be Premier League top scorer
While Allardyce was pleased with the nature of his side's performance, he has urged his squad to keep their feet on the ground ahead of the trip to his former club Newcastle United on Saturday.
"It should have been more than 2-0 against Cardiff on the chances, we were totally the dominant team," he said.
"We won't become overconfident [as] it's only game one. We were poor away from home last season, and I'll remind the lads that we had a great start last season then went to Swansea and got beat 3-0.
"You cannot ease off, you have to be 100 per cent focused and we have to try and improve our away form."
West Ham won only three league games away from Upton Park last season as they finished in 10th.
One of those rare away successes came at St James' Park, with Nolan the scorer against his former club in November's 1-0 win.
Allardyce, who managed Newcastle for eight months in the 2007-08 season, believes his former employers will be tough opposition, despite their 4-0 defeat at Manchester City on Monday.
"I've worked at St James' and in the first game of the season there the fans will lift the players. We have to be on top form," he continued.
"They had a poor performance on Monday but that was only brought about by circumstance and Man City being absolutely brilliant."
Allardyce will be without another former Newcastle man as striker Andy Carroll remains out with a heel injury.
However, Allardyce confirmed the 24-year-old has stepped up his recovery programme.
"Andy Carroll has had an injection to finally finish off the last bit of pain he's suffering in his heel, to shift inflammation," he said.
"It's completely repaired and we expect him to step up his training now and then we'll take it day by day."