The Welsh had built a 13-9 half-time lead at Murrayfield as they tried to keep their championship hopes alive, but tries from Tim Visser and Tommy Seymour helped the Scots to a 29-13 victory in Edinburgh.
However, Warburton has suggested that Wales owe it to their fans to bounce back when they host France in a fortnight's time.
He is quoted by the Western Telegraph as saying: "You can't win a championship losing two games so we know that's gone. For the rest of the campaign we are playing for pride and there were so many Welsh fans that went to Scotland, we noticed that on the bus on the way in and in the stadium.
"We appreciate that they spend a lot of time and money coming to watch us play so putting in a good performance for them in two weeks at home is what they deserve from us.
"We need to have a good, honest debrief of the game and we've been doing that over the last couple of weeks with players putting their hands up when they've made mistakes."
Wales last won the Six Nations in 2013, when they hammered England 30-3 in Cardiff on the final day to deny Stuart Lancaster's men a Grand Slam.