One-Way ANOVA:

F = MSG/MSE

S_p² = MSE

p-value = P(F_(I-1,N-I) > F_obs)

So if you increase S_p , F will decrease and as a result p-value will increase.

To see that click on the link below:

http://bcs.whfreeman.com/ips4e/cat_010/applets/anova.html