Chelsea went a man down after Victor Moses was shown a second yellow in the second half.
Despite their defeat, Fabregas insists Chelsea's campaign should still be classed as a success.
Chelsea looked off the pace and in the 29th minute Cahill was again required to clear off the line as Arsenal, on the counter-attack, went close through Welbeck.
And by the end of the season the Arsenal squad, as so often in the past, had been wrecked by injury.
The Italian said: "I think our start wasn't good enough".
Chelsea had been hot favourites to complete a remarkable debut season for manager Antonio Conte but was not at their clinical best as Arsenal, who finished fifth in the league, lifted the Cup for the third time in four seasons.
"I will be happy to stay for many years".
At Deepdale, Arsenal won 2-1 with goals from Aaron Ramsey and Olivier Giroud. "He put it on the plate for me to score the victor". With Arsenal losing out on the prestigious competition and Juventus hovering over the player, the board will pretty much have to throw the kitchen sink at him.
Armed police officers, an increasingly familiar sight in Britain, were a reassuring presence for followers of Arsenal and Chelsea in north London.
"We didn't care about what was said in the press or media, we just focused opn ourselves and we had a brilliant game today". Or will he leave the club on a high, beating a London rival to deliver a record-setting 7th FA Cup as a manager? "I don't mind facing criticism from supporters or from journalists, I will always try to answer it if I can, but once the game starts you should stand behind your team".
But I don't know whether this latest triumph affects the decision about whether he will leave the club this summer or sign a new contract. "This season for us was incredible because every single player gave all themselves to the work, they worked very hard and next season we must pay great attention to understand that we have to improve numbers in the squad and quality".