There were six tries in all for the 2003 World Cup winners, with centre Shontayne Hape and wing Chris Ashton getting two each, but it took four from the back line after the break to finally see off the dogged Georgians.
"We have a lot to improve on but you can only win each game in front of you and that's what we've done today," said Moody.
Tries from Delon Armitage, Manu Tuilagi and Ashton's brace made it 24 points without response in the second half, but with games against Romania and Scotland to come in Pool B Moody is looking for greater precision from his troops.
Hooker Dylan Hartley picked up a yellow card at the end of the second half and that gave Georgia the chance to grab their moment of glory as No 8 Dimitri Basilaia crashed over for a try which outside half Merab Kvirikashvili converted to bring his side back to within one score.
Kvirikashvili had earlier kicked a penalty and had he been more accurate with his boot - he missed with five of his seven attempts at goal - the Georgians might have got closer. His opposite number, Toby Flood, landed four conversions and a penalty.