Everton boss Moyes delighted with victory at Sunderland
There was an element of fortune in the result, with the winner careering into the net after a deflection from Andrew Johnson's arm, but Moyes believes the gutsy display spoke volumes for his side.
"It was always going to be a test. We got back late (from Italy) on Friday, travelled here yesterday and it can have its effects," he said.
"We could have gone to Fiorentina and played very well and lost 2-0 but we didn't, we played very badly.
"But the players showed great character, rolled up their sleeves and came away with three valuable points."
Sunderland manager Roy Keane admits defeat here, coupled with a disappointing 0-0 draw against bottom club Derby last time out, has hit hard but insists his side have both the quality and the self-belief to avoid relegation.
"I think there's five, six, seven teams down there fighting and we've said all along it might go to the last week or two.
"Our run-in is as difficult as anyone's which is why it's been disappointing this last week. We thought we had the opportunity to get three points at Derby and maybe a draw today," said the former Manchester United captain.
"Yes, the players are down and disappointed. The day they are not after a defeat will be a very sad day for the club.
"But we'll lick our wounds and get ready for another tough game (against Chelsea on Saturday).
"You have to keep believing, sometimes it's hard when you lose, but the spirit among the players is the least of my worries."