For 2x2 contingency tables, the "rule of thumb" for use of the Pearson's Chi Squared test is that:
1) All expected counts should be 5 or greater
2) If any expected counts are < 5, another test should be used, such as Fisher’s Exact Test.
In this case, you have a 2x2 with one cell that includes a value of 5, so Chi2 is an appropriate test to use. When you run the numbers, you see that both give statistically significant p values of <0.05.
For contingency tables larger than 2x2 the rule of thumb for using Pearson’s Chi Squared test is that “…no more than 20% of the expected counts are less than 5 and all individual expected counts are 1 or greater." (Yates, Moore & McCabe, 1999) So the "20% of cell" really applies to larger tables with more cells.