The Gunners remain sixth, but moved to within four points of arch-rivals Tottenham following the emphatic home win, which was sealed after four goals in the space of 10 minutes at the start of the second half.
"It was one game and we have three points which is important, but we have 15 games more to play. It is not about saying, 'today Arsenal are back', because we need more of these games and more of these points. Then I think we are back in the top four," the German said.
"There is enough time to get back in the top four, and a lot of points to play for. We need more of these games and then we are back in the first four.
"We must believe in our game and our players. After the two defeats against Manchester City and Chelsea, the atmosphere was not so good, but now we have won 5-1 and we look to the next game in the FA Cup."
Podolski chalked up his 11th goal against West Ham, as his impressive integration continues following an £11million summer move from Cologne.
The 27-year-old has, though, completed only a full 90 minutes in the Premier League once, and was again substituted.
Podolski has been deployed to great effect down the left, rather than a preferred central attacking role, but the German just gets on with the job at hand.
"When you are a footballer, you will not always play 90 minutes. Sometimes you are not happy, but this is not important for me," he said. "I do it well - I fight for Arsenal, for the club and team and this for me is important.
"We have a lot of great players. The coaches don't change the full-back or centre-back - you change always the positions in the front who create something. You have great players and it is not a problem when I am substituted."