The tourists looked set to claim victory and level the three-match series 1-1 only for Mike Harris to slot over a penalty with the last kick of the match.
Warburton told Sky Sports 2: "Speechless, absolutely demoralised to come so close.
"It's tough when they get momentum on the line-out and you have to stop it. Just gutted.
"It was very physical. We've showed a lot of character all season, it's just getting that vital win out here. I don't think you can get any closer than that to he honest. We have to pick ourselves up for next week."
Wales, who lost the first Test 27-19, play the third Test next Saturday.
Warburton added: "Now when we do take a lead we've got to capitalise on that and not let Australia back into game.
"We are desperate to get a result out here and this sill motivate the boys even more."
Australia captain David Pocock admitted his side had found the going hard.
"Credit to Wales, that was a tough game out there," he said.
Adam Jones could also not hide his frustration. "I'm obviously gutted," he said. "I think it's our own fault, with silly penalties we've given them a chance to get up our side of the field.
"It was a perfect chance for us to level the series tonight."
Wales caretaker coach Rob Howley admitted there were positives to take from the performance, but a lack of discipline at the death cost them.
"We were quite clinical when we had the ball and took our opportunities, but we just gave them one too many opportunities to get that kick at goal with discipline in certain areas," he said.
"I said to guys they could take great heart from their performance.
"I thought we defended well, the first half in particular when we needed to. We got off our line very quickly and made our tackles. In the second half through our defence we got our try and in just two minutes probably toward the end made a couple of wrong decisions and it's cost us the game.
"We'll be better for that but it's hard to swallow.
"In international rugby there's a fine margin between winning and loses. For 79 minutes and 20 seconds we've done a lot right."
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