The SPL side produced a dominant performance at Tulloch Caledonian Stadium with their Scottish First Division opponents showing little resistence.
Inverness captain Foran opened the scoring in 19 minutes with a 12-yard low drive and Rooney's shot from the edge of the area doubled the lead seven minutes before the interval.
Chris Hogg headed home from the edge of the area in 62 minutes before Rooney nodded in from close range three minutes later. Derek Lyle notched a consolation on 83 minutes but Foran completed the scoring two minutes later.
Northern Ireland international goalkeeper Jonny Tuffey was among six changes to the Caley Thistle starting line-up who lost 2-0 to Aberdeen last time out.
The shot-stopper, who was selected for his country's squad to face Scotland in the Carling Nations Cup on Wednesday, made his fourth start of the campaign as he was included ahead of Ryan Esson.
Starting debuts were given to new signings Aaron Doran and Chris Hogg, on loaned until the end of the season from Blackburn and Hibernian respectively, in place of Jonny Hayes and Grant Munro.
Meanwhile, Morton's line-up was unchanged from the side which beat Ross County 2-1 last Saturday.
The game went ahead after the pitch passed an 8am inspection following Inverness's home game against Dundee United on Wednesday night being postponed due to a waterlogged pitch.
Morton, who are sixth in the Scottish First Division, were unbeaten in the last six games but had lost to their SPL counterparts in their last seven meetings before today.
Looking to set the record straight, the visitors started brightly with Tuffey first to be tested when he palmed away Allan Jenkins' strike from the edge of the area after he was found by Brian Graham in the opening stages.
Caley Thistle started to show a threat with the recalled Lee Cox coming closest, while new boy Doran provided dangerous crosses.
Foran broke the deadlock for the Highlanders in 19 minutes when he hammered home from around 12 yards after the ball was laid off to him by Rooney.
Foran might have doubled his side's lead midway through the first half with a curling effort, but Morton keeper Colin Stewart was equal to it.
The Irishman came close again on the half-hour when his header from a Doran corner went just over.
The second finally came on 38 minutes when Rooney's effort went in off the post.
The striker's angled shot from the edge of the area was his 15th goal of the season and came after he was played in by Foran.
Morton manager Allan Moore made two half-time changes in a bid to stage a fightback with Carlo Monti and Peter Weatherson replacing Brian Graham and David O'Brien.
Monti could have levelled with a low angled shot from just outside the area but Tuffey blocked it.
Inverness might have added to their lead with headers from Eric Odhiambo and Ross Tokely but both were just off target.
Spanish attacking midfielder Dani Sanchez entered the fray for Odhiambo on the hour mark and supplied the cross for the third goal with his first touch.
Debutant Hogg powered home from the edge of the area to put Inverness 3-0 ahead.
Three minutes later Adam Rooney nodded in Caley Thistle's fourth from six yards when he connected with a Doran cross.
Morton keeper Colin Stewart denied Doran a goal when he punched away his 25-yard strike in 70 minutes.
Sanchez appeared to suffer an arm injury moments later and was replaced by David Proctor, from whose cross Rooney should have made it five.
The usually sure-footed frontman mis-kicked when he had a open goal from three yards out and Foran might also have increased the lead but his curling shot went inches wide.
Lyle's goal with seven minutes to go was no more than a consolation for Morton. The forward outmuscled Hogg and fired home from eight yards to make it 4-1.
Inverness responded in devastating fashion when Foran scored from a similar distance after being picked out with a long ball from Cox and sealed his side a place in the last eight.