The former Daily Mirror editor did congratulate Arsenal for their win, however, adding: "FT 2-1".
Ospina thwarted Costa and Ozil hit the post after cutting inside Cesar Azpilicueta, but just as in the 2014 final against Hull City, Ramsey's goal was to prove the victor.
In the fourth minute, Alexis Sanchez opened the scoring with a goal that could have been negated for a handball.
Ramsey also netted the victor in Arsenal's FA Cup victory over Hull City in May 2014 and he insisted the club's season should now be judged prosperous despite missing out on qualification for the Champions League for the first time in two decades. But after guiding the Gunners to an impressive triumph over Premier League champions Chelsea at Wembley, the team spirit and character shown was hugely impressive and testament to the players involved.More news: Tories criticise Jeremy Corbyn's record on terror laws
Per Mertesacker hardly had the ideal preparation for the FA Cup final.
"We must be proud we made it this far", Conte said.
Aaron Ramsey's header won the FA Cup for Arsenal as Chelsea missed out on the Double.
The Gunners were eliminated from the Champions League for the seventh successive season in the last 16. After a long, frustrating season which may yet prove to be his last in north London, the Frenchman secured the 10th major trophy of his tenure with a performance reminiscent of his best Arsenal teams.
Asked about Wenger's future, Arsenal's Danny Welbeck said: "The manager is his own man and he makes his own decision and the board will make the right decision so I can't comment on that".More news: Chelsea cancel Premier League victory parade after Manchester attack
It was the first time since the Dutchman's 2011-12 campaign that an Arsenal player reached the mark.
"I think this season has been hurt by some dip in March and I am convinced now, having looked back again at the season, we were hurt by the uncertainty about my future with the players", Wenger said.
Chelsea became the first English side to record 30 top-flight victories in a 38-game season and finished seven points clear of runners-up Tottenham.
Antonio Conte wants Chelsea to be "very cold" when they analyse their "incredible" season that ended on the sour note of FA Cup final defeat to Arsenal.
"I think we had an outstanding performance from the first minute onwards", Wenger said after Arsenal's record 13th success in the competition.More news: Arsenal deny Chelsea double as Ramsey seals FA Cup
Meanwhile, Arsenal win the cup again, and whatever happens with Wenger, he has delivered another trophy to the Emirates. I'm happy for the manager.
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