Celtic cruise into last 16 as Scottish Cup holders Rangers also progress
Japan forward Furuhashi matched his total for last season thanks to a first-half double in the fourth-round tie at Parkhead.
Aaron Mooy also scored twice, with David Turnbull on target as well.
Celtic went ahead from the penalty spot following a harsh VAR decision against former Hoops defender Efe Ambrose.
Bhoys manager Ange Postecoglou, who has made his opposition to VAR clear, had no desire to revisit the topic as he praised Celtic's approach to a potentially tricky cup tie.
"I thought the attitude of the boys was great," he said. "You go into these games with a bit of trepidation about how the boys will react and I thought they reacted well."
Morton could have broken the deadlock when Lewis Strapp was left one-on-one with Joe Hart only to shoot straight at the former England goalkeeper.
At the other end, Celtic's Liel Abada had a shot blocked and play continued.
But VAR eventually spotted that Abada's shot had come off the foot of Jack Baird then bounced up away from goal and brushed Ambrose's arm, which was slightly away from his body.
That was enough for Napier to decide on-field referee Kevin Clancy had made a "clear and obvious" error.
Having looked at the pitchside monitor, Clancy awarded Celtic a penalty.
"I don't know how Efe's meant to get his hand out of the road," said Morton manager Dougie Imrie. "It takes a deflection and it's so close.
"I just don't know how they can give a penalty."
Mooy sent Brian Schwake the wrong way from the spot in the 18th minute. Three minutes later Celtic were 2-0 ahead when Ambrose took a heavy touch on the edge of his box and Abada played in Furuhashi
Celtic went further ahead in the 42nd minute when Furuhashi laid the ball back for Turnbull, who let fly with a first-time shot from 22 yards.
Furuhashi made it 4-0 in stoppage time after Grant Gillespie had taken a fresh-air swipe at Abada's cutback.
Morton denied Celtic a second-half goal until six minutes from time when Mooy finished from close range.
Rangers made it nine games unbeaten under manager Michael Beale with full-back Borna Barisic's thunderous shot on the stroke of half-time, after a Fashion Sakala effort came back off the post, proving the difference at St Johnstone.
"I'm delighted to get through," said Beale. "We've had three games in six days. Fantastic mentality from the group."
Premiership high-flyers Livingston avoided a Cup upset by coming from behind to win 3-1 at League Two Stenhousemuir.
Top-flight Kilmarnock beat League Two leaders Dumbarton 1-0, and Dundee United also had a difficult afternoon before seeing off Stirling University 3-0.
Hearts, last season's losing finalists, make the short trip to Hibernian for an Edinburgh derby on Sunday.