I'd be interested to see if there are any real ways to do this, but I think the answer is "no" unless your landlord accepts credit card payments.
The obvious method of taking a cash advance from the card to pay the rent won't work, because you don't earn rewards on cash advances (at least not on any card I've ever seen), usually interest starts accumulating immediately on cash advances (no grace period) which would eat up the "savings" from rewards, and usually there is a percentage fee (3-5% for cash advances) that would also eat up the rewards.
You might be able to get your landlord to start accepting credit cards in exchange for a 5% higher rent. With the new card readers from Square for example it is very easy and convenient.
The landlord will have to pay merchant fees for taking cards so you should expect to pay a little more to cover that.
I take credit cards for rent on our short term rental. Visa, MC, and Discover have no issue at all with it. American Express has some special rules so it is hard to do it with them, though not impossible.