Griffiths replaced Odsonne Edouard in the 97th minute after Hearts had recovered from two goals down to force extra-time at an empty Hampden.
The Scotland striker pounced just a few minutes later to put the Hoops 3-2 ahead, only for Josh Ginnelly to equalise for Hearts and force the game into penalties.
Griffiths then scored from the spot as Celtic claimed a 4-3 win in the shoot-out and the 30-year-old from Leith was delighted to get one over at his former derby rivals.
"I've been humiliated in a cup final against Hearts so I wanted to try and get revenge," he told Premier Sports.
"It might not be a 5-1 scoreline but it's another medal round the neck and I'm absolutely buzzing."
Griffiths also demanded respect for Celtic's historic quadruple treble, insisting: "It's hard for teams to do one treble.
"We've done four on the bounce now and I think we need to get the credit we deserve.
"We've jumped every hurdle that's been in our path and it's something I don't think we'll ever see again in British football.
"We've been criticised and rightly so because performances haven't been great, but hopefully that's a turning point now.
"We've got a few games that we can hopefully pick up three points and we look forward to the big match [against Rangers] at the start of January."