Warburton's side trailed 10-6 at the break, but Shane Williams scored a crucial try to secure the win.
"I would like to dedicate the win today to the families of the miners back home," said flanker Warburton. "Our thoughts are with them."
Coach Warren Gatland said: "I thought we showed some great character."
It was Wales' first World Cup win over the islanders in three attempts, having lost to them in Cardiff at the 1991 and 1999 tournaments.
Victory also followed Wales' opening 17-16 defeat by reigning champions South Africa.
Gatland, coaching Wales in his home town of Hamilton, added: "We were under pressure at half-time but the boys dug deep.
"We've worked really hard the last few months and you've got to show people how much it means to us to get out of this and get a W [win] on the board.
"I thought in the last 15-20 minutes Samoa's conditioning probably told and they looked to tire a bit and weren't quite as effective in the second part of the second half as they were in the first.
"I thought we defended exceptionally well for most of the game."
Flanker Warburton praised Leigh Halfpenny, who replaced shoulder injury victim James Hook at the break. Wales also lost Dan Lydiate to an ankle injury.
"Both sides didn't disappoint, we knew it was going to be an immensely tough battle," said Warburton.
"Samoa are very physical and it was a tough game for us.
"We wanted to keep the ball in play as much as we could and back our fitness levels towards the end of the match.
"Leigh [Halfpenny] came on and added some great momentum to the side and Shane finished it off as we've seen him do a million times before.
"If our backs get a sniff they're pretty handy."
