Rovers are among six sides yet to get off the mark in the Premier League this season, although they have only played one game.
A 2-0 reversal at Ewood Park against Manchester City on the opening day remains their only tie this term after they were spared from the midweek round of fixtures.
While there is no need to panic just yet, Allardyce feels that another setback at Sunderland on Saturday would leave his team playing catch up early on.
"We had no midweek game, so all of a sudden the pressure is there to secure the first points as quickly as you can," he said.
"We've got to do that at Sunderland - we've got to go and get a positive result, and come back with some points on the board.
"There's only been two games, but some teams will have played three when it comes up this weekend.
"It starts after two or three games now - its not like six, eight or 10, it's after two or three."
ResultsAllardyce is keen to keep his players away from any media talk of how vital early games can be, but the experienced coach knows points are a priceless commodity regardless of where you stand in the table.
"Looking at what has been said about Man United and Liverpool, already Chelsea are the favourites, people are saying they are going to win the title," he said.
"In the same way they are writing off Portsmouth already, but that's what you get.
"You tell your players not to read the papers, but it is impossible not to. So we've got to get results as quick as we can, get the points on the board.
"When you lose a home game it's always difficult to make up, but you have got to make it up somewhere, and now that's going to have to be away from home. So let's hope we start picking up some points at Sunderland."