Victor Anichebe came on as a late substitute and scored three minutes from time to lift Everton over West Brom Albion.
Victor Anichebe makes most of his time for Everton
Victor Anichebe climbed off the bench to net the only goal of the game in the 87th minute to help Everton claim their fourth away win of the campaign and help them leapfrog Albion in the table.
It also extended the Toffees' unbeaten run to four matches in the Premier League, and Distin has urged his side to keep that sequence going.
"It's a good run, we have to carry on," he said on Sky Sports News. "We have to turn the draws into victories and that would be even better."
Distin was named man of the match after helping Everton keep a clean sheet, but the Frenchman insists the plaudits for that do not just belong with the back four.
"It's not only the defence, it's the team," he said.
"If you only expect the back four to defend you are never going to succeed and keep a clean sheet. You start from the front and everybody worked hard today, and at the end we've got three points and a clean sheet, which is perfect."
Everton match-winner Anichebe, who has been troubled with injury problems, said after scoring his first league goal of the season: "I've been waiting patiently since the start of the season, I had a good pre-season but unfortunately I went away with Nigeria and got injured.
"He [manager David Moyes] said, 'Go on, you've worked hard to get back here, just go on and help the team', which is what I did."
West Brom, the league's lowest scorers at home with just seven goals in 10 matches, created more opportunities but failed to register a shot on American Tim Howard's goal, with Peter Odemwingie wasting their best chance when he volleyed over the bar in the 47th minute.
Shane Long, Odemwingie's strike partner, had earlier headed over another decent opportunity in the 18th minute.
"Today we've given three very big points to our rivals," said Roy Hodgson, the Baggies manager. "They did well enough to deserve a draw but must be feeling very grateful that we gifted them a goal three minutes from time."