Arsenal ended their nine-year drought without a trophy by edging Premier League rivals Hull City 3-2 after extra-time in a dramatic FA Cup final at Wembley Stadium.
Arsene Wenger's men had trailed 2-0 inside the opening eight minutes but goals from Santi Cazorla and Laurent Koscielny restored parity, before Aaron Ramsey netted the match winner in the 108th minute.
It ensured Arsenal ended a difficult season on a winning note after they slipped out of the Premier League title race to finish fourth, having topped the standings for 128 days of the season, and failed to make a significant impact in the UEFA Champions League (Round of 16) and Capital One Cup (fourth round).
But Winterburn, who made 440 league appearances for the London club between 1987 and 2000, insists Arsenal's long-suffering fans should be proud of the team's achievements this season.
"It is a success (Arsenal's season) and don't let anyone else tell you differently. Because as a player you set out to win every competition you're in," Winterburn told Perform after the match.
"You start with the Premier League, you go to the Champions League if you're looking to win it, then you go to the FA Cup.
"If you don't win the first two, but you win the FA Cup then it's a successful season, along with finishing fourth. Fourth was the achievement, the FA Cup was a success.
"And no-one else should tell you it's not success, because there are a lot of teams out there who would be proud to be lifting the FA Cup."
While Winterburn believes ending the drought is important, he wants Wenger to maintain Arsenal's momentum by strengthening the squad during pre-season, with out-of-contract defender Bacary Sagna expected to leave on a free transfer.
"I would like to see him spend some money to really restrengthen this team," the 50-year-old said.
"I don't see any reason why he wouldn't restrengthen and see if this team can really push on and challenge for a Premier League title over the next couple of seasons."
