They did not play a part in a spate of midweek games, but with there having been no draws so far out of 16 league fixtures, Allardyce knows his team need to get up and running as soon as possible.
"We had no midweek game, so all of a sudden the pressure is there to secure the first points as quickly as you can," Allardyce 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."
Allardyce suggested that with every Premier League team currently on either zero, three or six points, the pressure to keep pace with rivals could be felt across the entire division.
"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."
Sunderland already have a win under their belts after clinching victory at Bolton on the opening day, and they took the lead against Chelsea on Tuesday night before going down 3-1.
Allardyce - who both played and coached at Sunderland - admitted he was impressed with the direction the Black Cats are going in and identified their new two-goal striker Darren Bent as the man his team would need to watch most closely.
"I worked there twice, once as a player and once as an academy manager for Peter Reid, so I'm well aware of the passion in and around Sunderland," Allardyce said.
"They are the top dogs up there now in the north east, the only side up there in the Premier League. They have a new owner, a great chairman and a very good manager to boot now.
"They are trying to put an ambitious club together to make sure it doesn't struggle in the league as it has done in the last two years, but we've got to go and spoil the party.
"They made a great effort against Chelsea on Tuesday night, but were overcome by outstanding ability.
"We have to go and match them and stop Darren Bent from scoring. He only had half a chance on Tuesday and put it in the back of the net, and scored a good header at Bolton, so we'll have to take care and look after him."
In terms of his own strike force, Allardyce acknowledged the need for a more ruthless edge in front of goal after the team squandered a host of opportunities against Manchester City.
He said: "You're always happy when you are creating chances on goal, but we had 20 attempts with 10 on target on Saturday, which is a huge amount in a Premier League game, and no goals at the end of that is really disappointing.
"We have to continue making the chances but be more clinical in front of goal."
Allardyce's new signing Michel Salgado has joined up with the squad but will not have received clearance in time to make his debut at the Stadium of Light on Saturday.
Sunderland v Blackburn. Click here to bet.