After Diomansy Kamara opened the scoring for the visitors, David Graham's strike and an Andy Kirk penalty put the home side ahead before Morten Rasmussen levelled two minutes before the interval.
The Fifers' resistance crumbled in the second half after defender Callum Woods scored an own goal before Keane, on for Marc Crosas, scored his first goal for the club from the spot after he had been felled by Pars' skipper Austin McCann.
Celtic now turn their attention back to the league where they remain 10 points behind their Old Firm rivals and Mowbray is hoping for a big crowd on Wednesday night to see Keane.
"I hope Robbie is fired up for Wednesday, that's why he only played 45 today," he said.
"He was always going to be playing 90 minutes on Wednesday, or at least starting the game.
"I spoke to Robbie during the week (after the Kilmarnock game).
"Three 90 minutes in seven days for the boy is probably asking for trouble injury-wise, and I think he agreed.
"But I am delighted Robbie is here because he adds that finesse in the final third that sometimes makes that difference in games.
"There was some moments of magic today when he sat players on their backside in the box, in tight areas.
"He has got that quality - and he can take a penalty as we saw.
"They are all tough games, Wednesday will be a hard game but hopefully the extra expectation and tension lifts the group.
"Hopefully our supporters will come out in numbers to support the team and see Robbie perform."
Mowbray believes Keane is already impressing his new team mates but not in a way usually associated with superstar footballers.
"Robbie is a very humble guy who doesn't carry a superstar ego around with him," the Celtic manager said.
"He just wants to train and play and pass his advice on
"He is just a good guy to have around and he loves football.
"He will have many chances to play at Celtic Park and I hope Wednesday will be the first of many successful games for him."
Rasmussen cheekily tried to claim Woods' own goal to give himself a brace.
He said: "I was going after the ball and I think I got a little touch as did the defender, so I will take the goal and that makes it two today!
"But it was important that we won and got through to the next round."
Pars manager Jim McIntyre will look again at the second-half penalty Celtic were awarded.
It looked to be a strong but fair tackle on Keane but referee Charlie Richmond pointed to the spot allowing the Irishman to score his first Celtic goal.
McIntyre said: "Austin is claiming he got the ball so I would need to see it again. It was difficult to see.
"But I said to the players before the game that they would need to believe that we could get a result and if they didn't then they could end up on the end of a hiding.
"So I was very pleased by the way we approached the game and I was really pleased by the way we came back after losing the goal."
Source: Team Talk
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