Hamish Rutherford, the New Zealand batsman, gave Essex a flying start in pursuit of a target of 149 with 30 off 17 balls before Greg Smith and Owais Shah shared a second wicket partnership of 98 in 12 overs to enable them to cruise home with seven balls to spare.
Their fourth win puts them alongside Hampshire at the top of the South Division although they have an inferior run rate and have played two games more.
Ponting, who had scored only 35 runs in his three previous innings in the competition with a highest score of 19, found himself at the crease in the second over after Steve Davies had got a leading edge to his first ball from Graham Napier and ballooned it back to the bowler.
Jason Roy had already hit two fours off Shaun Tait in the first over and he struck two more off David Masters in the third but he and Ponting had added only 43 in six overs when Roy was run out for 30 by Ryan ten Doeschate's throw from midwicket.
Ponting and Glenn Maxwell were no quicker in a third wicket stand of 45 in seven overs but the former Australia captain looked positively skittish as he scooped ten Doeschate to fine leg for four and drove Reece Topley over long off to reach his fifty off 40 balls.
He had lost Maxwell when he was brilliantly caught one-handed by Tim Phillips on the midwicket boundary off ten Doeschate and it was Phillips who held the catch at long on to get rid of Ponting off Topley after he had made his 65 off 54 balls with six fours and a six.
It had been another brilliant performance by the 38-year-old who last week finished his first-class career with an unbeaten 169.
Essex did well to restrict Surrey to 148 for six and did even better with the bat, Rutherford setting the pace with two fours and two sixes before he was well caught at deep backward square leg off Jon Lewis.
Then Smith took over, accelerating to 62 off 42 balls with four fours and three sixes, and Shah gave him all the support he needed with 46 off 45 balls, including two fours and a six.