In a disappointing encounter, West Ham had the better chances in a tight contest, but could not find a way to goal, with Kevin Nolan missing the best opportunity to net the winner against his former club.
Allardyce was happy with his side's performance, but felt they should have been rewarded with all three points after restricting Newcastle to no real goalscoring opportunities.
"We are not disappointed, but we know we should have won this game and can only blame ourselves for our lack of finishing quality," Allardyce told Sky Sports.
"It is not that often when you play away from home and create three or four clear-cut goalscoring chances - one for Stewart Downing, one for Modibo Maiga and one for Kevin Nolan, but unfortunately our heading ability was not up to scratch.
"The point is well received, but we all know we should have had all three today.
"It is very satisfying to come away from home and get a point, so four points from two games is a good start."