Joe Ledley put Celtic ahead in the 40th minute, and although former Celt Cillian Sheridan levelled two minutes after the break, it was a mere catalyst to a dominant second half for the home side.
Matthews arrowed in a shot from 25 yards and substitute Anthony Stokes grabbed a third Celtic goal. Matthews sealed a convincing win from close range to take the champions 15 points clear of Inverness at the summit.
Second-placed Inverness stumbled to a 2-1 defeat at League Cup finalists St Mirren, the team that denied Celtic a place in the Hampden showpiece.
Esmael Goncalves scored on his home debut for St Mirren midway through the first half, before Inverness drew level on the stroke of half-time when Billy McKay netted his 20th league goal of the season, this one from the penalty spot after Conor Newton felled Richie Foran inside the box.
Steven Thompson's 82nd-minute header earned the Buddies all three points.
John Sutton's 86th-minute goal broke Dundee's resilience as Hearts beat the league's bottom-placed side 1-0 at Tynecastle.
Arvydas Novikovas let a cross-field pass slip under his foot but recovered and sent in a deep curling cross from the left that was powered into the net by Sutton at the far post.
Ivan Sproule scored against his old team in the 36th minute to earn Ross County a deserved 1-0 home victory over Hibernian.
St Johnstone were 3-1 winners at home against Aberdeen, who paid the price for some hesitant defending.
Gregory Tade opened the scoring for the hosts in the 33rd minute after a mistake from stand-in centre-back Isaac Osbourne but Niall McGinn's superb volley brought the Dons level a minute before half-time.
Rowan Vine hooked home after a 68th-minute corner and Dave Mackay added a third Saints goal soon afterwards for the Perth side.
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