Reds goalkeeper Pepe Reina was then called into action twice to keep the Italians at bay.
Di Natale had one more chance before the half-time whistle but the dangerman could only lash his shot narrowly over the crossbar on the angle.
After an impressive first-half display, Liverpool were undone 35 seconds into the second half when Di Natale fired in from Andrea Lazzari's cross.
However, the Italians continued to pile on the pressure and were rewarded in rather fortunate circumstances in the 70th minute when Coates headed into his own net.
It got worse for the Merseysiders just two minutes later as Giovanni Pasquale fired in from the edge of the area.
Despite losing two goals in as many minutes, Liverpool rallied and cut the deficit soon after thanks to Suarez's free kick.
The Uruguayan had two chances to level the scores late on but he was kept out by a rugged Udinese defence.