The Hoops broke the deadlock at Hampden Park after just 10 minutes as a poor clearance from Charles Dunne was controlled by Callum McGregor and rifled into the top corner of the net from just outside the box.
Brendan Rodgers's charges then doubled their advantage on 25 minutes through Olivier Ntcham, who was on hand to connect with Moussa Dembele's probing ball on the edge of the box before skidding a low effort beyond the reach of Trevor Carson.
Motherwell had little response against Celtic's domination and almost conceded a third as Tom Rogic tried his luck from around 12 yards out, but goalkeeper Carson was equal to the Australian's 35th-minute effort.
Indeed, Motherwell grew increasingly frustrated as Richard Tait and Liam Grimshaw were booked for late challenges on the opposition players before the whistle blew on a deflating first half for the Fir Park side.
The second half brought about an improvement from the Steelmen, Main testing Bhoys keeper Craig Gordon with a vicious strike that was tipped over the crossbar at the near post.
Both sides enjoyed several chances to add more goals in the latter stages and Well finished strongly, with Gael Bigirimana hitting the woodwork with a free kick.
However, the Scottish Premiership champions and Scottish League Cup holders could not be denied a historic win as they added the Scottish Cup to their trophy cabinet in a clean sweep of domestic silverware.
CELTIC (4-2-3-1): Gordon; Lustig, Boyata, Ajer (Simunovic 76'), Tierney; Ntcham, Brown; Forrest (Sinclair 90'), Rogic (Armstrong 73'), McGregor; Dembele
MOTHERWELL (3-5-2): Carson; Kipre, Aldred, Dunne; Cadden, Campbell (Frear 78'), McHugh (Bigirimana 56'), Grimshaw, Tait; Main, Bowman