Higdon headed home a 17th-minute corner after a slow start from the visitors, who featured six changes to the team that lost to Rangers on Sunday.
Law scored a glorious second four minutes before the break when he cut in from the left and beat two men before curling the ball into the corner of the net from 18 yards.
Tom Hateley fired the third just after the restart and Lawless capped an assured debut with an emphatic finish.
The bonus for Stuart McCall was that he rested Darren Randolph, Stephen Craigan, Jamie Murphy, Steve Jennings and Stevie Hammell ahead of Saturday's Clydesdale Bank Premier League clash at Dunfermline.
McCall handed a first start to left-back Jordan Halsman, while Jonathan Page, Ross Forbes and Bob McHugh also came in to form a youthful team.
Lee Hollis made his second appearance for Motherwell and was the busier of the goalkeepers in the opening spell.
The former Airdrie player got down to hold well from Stephen McDonald's header and John Neill's long-range drive, and impressively dribbled past two men in the box after Shaun Hutchinson's slightly short passback.
Motherwell fans were coming in throughout the opening stages after being held up in traffic gridlock outside Broadwood but they did not miss much from their team, who under-hit several passes on the soggy surface in the early period.
But they made the breakthrough when former St Mirren striker Higdon beat Clyde goalkeeper John Hutchinson to an inswinging corner from Forbes and headed home from six yards.
Hutchinson went some way to atoning when he pushed wide Keith Lasley's deflected effort, which bounced off the skiddy surface.
But the Clyde keeper could do nothing about Law's composed finish as the midfielder celebrated his first goal since his summer move from Rotherham.
Motherwell came out after the restart determined to finish off their Irn-Bru Third Division opponents.
The Clyde goalkeeper tipped over Higdon's looping volley and McHugh inadvertently blocked team-mate Hutchinson's header crossing the line.
Paul Hay saw a shot deflected wide after some uncertainty across the Motherwell defence but the game went well beyond the hosts in the 51st minute when Hateley sent a powerful left-foot strike high into the far corner after breaking into the box.
McCall quickly took the opportunity to give Law, Higdon and Lasley a rest with Chris Humphrey, Gary Smith and Stuart Carswell coming on.
Motherwell continued to dominate and McHugh and Humphrey both had powerful shots pushed wide by Clyde goalkeeper Hutchinson before Smith curled just wide from a difficult angle. McHugh shot over after being set up by Lawless, who rounded off the scoring in the 85th minute when he crashed a volley in off the underside of the bar from 15 yards after McHugh's effort had been blocked.
Defeat would not come as a surprise for Jim Duffy's men but they suffered a further blow when striker McDonald was stretchered off.