An extension of Chernoff bound
What you want is the generalized Chernoff bound, which only assumes $P(\bigwedge_{i\in S} X_{i}) \leq p^{|S|}$ for any subset S of variable indices. The latter follows from your assumption, since for $S=\{i_1,\ldots,i_{|S|}\}$, $$P(\bigwedge_{i\in S} X_{i}) = P(X_{i_1} = 1)P(X_{i_2}=1|X_{i_1}=1)\cdots P(X_{i_{|S|}}=1|X_{i_1},...,X_{i_{|S|-1}}=1)\leq p^{|S|}$$
Impagliazzo and Kabanets recently gave an alternative proof of the Chernoff bound, including the generalized one. In their paper you can find all the appropriate references to previous work:
http://www.cs.sfu.ca/~kabanets/papers/RANDOM2010.pdf