Question related to Odd ratio and P value



I have a question related to odd ratio and statistical significant that I'd like to get your help. I have two groups A and B . Its odd ratio is 1.09. Its Z score is 57.215 and the P value is less than 0.001. Since it has a small P value then there is a statistical significant between groups A and B. But my question to you is that why an odd ratio is almost identical to one result in a statistical significant different between two group A and B because I thought that since the odd ratio is almost equaled to one then there would be resulted in NO statistical significant between two groups A and B because their odd equaled to one. Please correct me if I am wrong because I am puzzle by this odd ratio and statistical significant.
Thanks in advance for helping.