The Spain midfielder set up Ivan Rakitic for the opening goal and was on the pitch for Luis Suarez’s winner, which arrived in response to Alvaro Morata’s equaliser for Juve.
Neymar added the gloss to the victory by making it 3-1 in stoppage time and the Catalans could finally celebrate adding the European Cup to the Primera Division and Copa del Rey already secured this season.
Iniesta told TVE: “We couldn’t have asked for more. To have won three titles and to end the campaign lifting the Champions League is incredible.
“It’s very difficult to win three titles in any given season and to have done it again is spectacular.
“Both teams had their chances.
“It would have calmed us down to have scored one more goal in the first half but nevertheless we got it done.
“To have won the treble twice in six years is something that will be very difficult for a team to repeat.
“It’s an incredible achievement for everyone, the players, for Barca and for our fans.”
The Catalans struck after only four minutes, Rakitic converting Iniesta’s assist to cap a remarkable 16-pass manoeuvre.
Lionel Messi did the leg work for Suarez’s close-range finish – the Argentinian’s shot was saved by Gianluigi Buffon before the Uruguayan pounced – and substitute Pedro slipped Neymar in for a late third.