Captain Sam Warburton says Wales "owed" coach Warren Gatland victory against Scotland.
The Welsh have never lost to the Scots during Gatland's reign and Warburton said that statistic spurred them on.
"We didn't want to be the team who were the first ones to lose under Warren [against Scotland]," he said.
"We owed it to Warren because we let ourselves down as well as the coaches and the fans last week [against England]."
Wales fought back from 10-3 down to race into a 26-16 lead with 10 minutes to go thanks to tries by Rhys Webb and Jonathan Davies.
Jon Welsh scored a late try for Scotland, but referee Glen Jackson did not allow a re-start although the clock indicated several seconds left.
"Warren's pleased," said Warburton.
"He said why couldn't we have done it slightly easier because he was stressing out a little bit. I don't blame him.
"That was definitely a game where our defence won us the match.
"There were times at the end of the first half when we held them out and in the second half and our set piece defence got slowly better as the game went on.
"It's such a tough place to come. That was a massive Test match. I'm battered and bruised.
"I'll definitely have an early night tonight with a bar of fruit and nut and get in bed nice and early."
Wales are now fourth in the Six Nations table after two matches, level on points with third placed France who they play next in Paris on Saturday, 28 February.
Warburton said his side had put "a lot of pressure" on themselves to beat Scotland.
"It would take a mathematical miracle to win the Championship with only three wins, so we know we had to win today," said the 2013 British and Irish Lions skipper.
"Every game becomes a must win game now if we have any chance of winning the championship.
"If we'd lost today I think you're as good as dead and buried."
"There's still plenty to work on. That was by no means a complete performance.
"Everything really, the set piece, our kicking game, it's a good place to be because we won the game and there's still plenty to work on.
"We've got two weeks now to iron out the creases before our next fixture."
Wales have won the title before after losing their opening match of the tournament, in 2013.
Warburton says although the win in Scotland keeps their championship hopes alive it does not change anything.
"We are still in the same situation because every game is a must win. We slip up once now and that's the championship gone," he said.
"When this team's put under pressure I think that's when we play best."