An Adam Hughes try helped the visitors to a 10-0 lead before Sean Reidy's score cut the margin to 10-7 by half-time at Kingspan Stadium.
After a Carl Meyer try extended the Dragons' lead, Ulster pressure were awarded a penalty try but the Welsh region looked set to hold on before Jackson's late kick.
"You can't ask for much more, I was quite proud to be involved with the guys," said Jones.
"To be leading after 77 minutes away in Ulster, it's very difficult to take losing after that.
"It's no good feeling sorry for ourselves, we've got to turn it round and face a Connacht side who have had one win in the last four but those defeats have been very tight and they gave the Scarlets a bit of a hiding.
"We feel we're inches away from being a mid-table team, a couple of inches here and there and we would have beaten one of the top teams away from home."
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