Jones's drop-goal on 75 minutes made it 15-14 to Wales but Ronan O'Gara answered with his own three minutes later.
"They were really on me so I really had to get the ball up - it was ugly but it flew straight and that's all that matters," he said.
"But it was 90 seconds too early - we gave them another shot.
"My heart goes out to Stephen - the game also meant so much to him."
O'Gara was quick to commiserate with Jones after the final whistle and swapped shirts with his opposite number.
"It's so unfair the pressure all comes down to the kickers," he said. "I just wanted to console with him and congratulate him on a good game. I know Stephen well enough that I might get my shirt back for a few quid!"
Grand Slam is all down to groundwork - Kidney
It was a remarkable triumph for Kidney, who was taking charge of the team in the Six Nations for the first time, but he was keen to stress the debt he owed to his predecessor Eddie O'Sullivan and those involved in grass-roots rugby.
"This is all due to the groundwork done by Eddie and all the coaching team, and all that's done in the provinces, and more so in the schools, as they're the ones who enrich the kids," he said.
"Brian would say this is far from a one-man team - we have 30-odd people who make this team.
"After 80 minutes, you take what's there - some days it swings for you, some it doesn't."
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