Celtic have now won all three domestic trophies on offer -- the Scottish Premiership, Scottish Cup and Scottish League Cup -- for the third consecutive year.
Celtic won the League Cup in December, beating Aberdeen 1-0 in the final, and sealed their eighth consecutive league title earlier this month.
"You'll never see this again," Celtic boss Neil Lennon, who succeeded Brendan Rodgers in February, told the BBC. "In footballing parlance, it's impossible (to put this into words). That's nine trophies out of nine, it's remarkable."
Ryan Edwards gave Celtic a scare with a goal for Hearts in the 52nd minute but French striker Odsonne Edouard grabbed a brace, the second a penalty, to give Lennon's side victory and a place in history.
After the match, Celtic chief executive Peter Lawwell confirmed that Lennon had been offered the job on a permanent basis, having taken over on an interim basis following Brendan Rodgers' switch to Leicester City.
Lawwell said: "Neil, in our moment of need, stood up to the plate. Over the next few days, we will work out the details.
"We're absolutely delighted. Fantastic day, fantastic achievement today - probably never to be repeated.
"At this point in terms of looking at the squad, there's nobody better that I've worked with who's got a better eye for a player. I think it's better to do that [offer him the job] tonight to just kill the speculation."