The Parkhead side had found the national stadium a tricky venue over the last two seasons, falling at the hands of Kilmarnock, Hearts and, most recently, St Mirren in the semi-final of the Scottish League Cup.
Mulgrew told reporters after the game: "We don't get caught up in the Hampden record too much. We'll let other people do the talking.
|6/4||Gary Hooper is 6/4 with BetVictor to join West Ham from Celtic this summer
The Scotland international has been moved positions throughout the season by manager Neil Lennon and only found out that he would be at the heart of the defence just before the game.
"I found out I'd be playing centre-half about an hour-and-a-half before the match," Mulgrew revealed. "It's somewhere I've played a lot, especially last year."
"I was happy to be involved in a cup final, to get a clean sheet and to win the game. I don't know what the manager's thinking was in the lead-up; you just prepare yourself to be ready for any scenario."
The former Wolves defender noted that stopping on-loan Hibs striker Leigh Griffiths was key to victory, observing: "We'd obviously played against Griffiths before. We knew what his strengths were and that he can score goals, so it was up to us to match that.
"More importantly, the onus was on us to win the game and score goals. We've got really good players up top and at the back."