Two ways to beat Miracle:

1. Put out about 20-30 damage out on the table and wear down its health to where it's about 2 turns away from dying, then drop a bunch of high damage creatures out at once to kill it in one blow (Parallel Universe on its biggest dragon works, or on your own biggest creature if you've got 20+ damage creatures also). Alternately, if you have a lot of direct damage (firebolt/drain) you may be able to use that, but Miracle usually puts out jade shield so that's not reliable at all.

2. Boost your damage output until you're doing 100+ damage per turn. Preferably, 200+. Dive/Parallel Universe works beautifully here, although every other level 6 deck has creature control that will make a dive/PU deck pretty crappy at anything other than farming Miracle specifically. (But if you don't have any Miracle spells of your own, putting together a dive/PU and hunting for Miracle matches is probably about the fastest way to get them.)