Beattie, making his debut for the Merseysiders following his £6.5million move from Southampton, laid on the opening goal for Leon Osman in the 16th minute.
Although Everton doubled their lead within two minutes through James McFadden, Beattie's evening gradually went downhill thereafter.
He was booked in the 36th minute after Argyle had pulled a goal back through Bjarni Gudjonsson for clattering into Argyle full-back Paul Connolly, and then missed an open goal when it looked easier to score, spooning over a fine low driven cross from Alessandro Pistone.
He could make little headway against Argyle's solid centre-back pairing of Mathias Doumbe and Graham Coughlan and was eventually substituted midway through the second half.
After withstanding serious pressure from the home side, Everton eventually made the game safe six minutes from time when substitute Nick Chadwick got the luck of the bounce before breaking away and converting with aplomb.
Nevertheless, Everton manager David Moyes was pleased with his new signing's first performance in his new colours.
"I always thought it would be an ideal game for him to kick off," said Moyes. "I told him that he would play for roughly an hour. His match-fitness is not quite where I'd like it to be but his contribution was very good in the first half.
Everton took the lead when Osman played a one-two with Beattie and coolly chipped Romain Larrieu and doubled the margin in their very next attack when Argyle failed to deal with a long throw into the box.
Despite having several chances to clear the ball in a penalty-box scramble, they somehow allowed McFadden to come up with the ball and smack it home from unmissable range.
Argyle midfielder Lee Hodges, set up by Gudjonsson, brought a flying save out of Everton goalkeeper Richard Wright with a dipping left-foot volley, and Gudjonsson himself let fly with a threatening long-range shot that was deflected for a corner before Plymouth clawed a goal back.
Goalkeeper Larrieu's drop-out was headed on by Micky Evans into the path of Gudjonsson, whose perfect control allowed him to be able to flick the ball past Richard Wright with the outside of his right boot.
The Championship side kept the pressure on their Premiership visitors right the way through a pulsating second half until Chadwick won the ball in the centre-circle and strode forward from halfway before calmly placing the ball past Larrieu.
Moyes paid tribute to the battle put up by the Devon side. "A lot of credit goes to Plymouth," he said. "Their supporters got right behind the team, and they competed and tried to get balls into our box as often as they could and made it difficult.
Argyle boss Bobby Williamson said he was proud of his players, although disappointed to conced two goals so early.
"The usual things go through your head 'If we capitulate, it could be a lot more; if we dig in and hopefully get that break, we can back into the game.'
"Thankfully it was the latter that happened and we certainly were back into the game.
"I've never had many opportunities to feel not proud of these guys. They work very, very hard for each other. Unfortunately for us, we've lost the game and are out of the cup.
"The first goal was a decent move from them. They cut us open slightly and got a lucky ricochet which put it into the path of their player, and he finished it very well.
"It was a long throw into our box which we didn't defend very well. I think Mat Doumbe had three opportunities to get something on it, and he did get something on it, but not enough."
Williamson also felt Argyle should have had a penalty when Paul Connolly was fouled by McFadden with the Pilgrims pressing for an equaliser.
"I thought it was a penalty, but I haven't had a TV replay to look at. Some you get, some you don't. I felt most of the 50-50 decisions went to the so-called bigger club. When the referee and his assistants had any doubt at all, there was only going to be one decision made.
"I'm not going to go down that road. We gave a good account of ourselves. Unfortunately we need breaks to go in your favour and I don't think they did. Even the third goal, I felt was a handball on the halfway line.
"We knew we could have beaten Everton if we got the breaks. We did trouble them at times, but we just didn't get the breaks."
MAN OF THE MATCH: Marcus Bent (Everton) - Seemed to profit from the introduction of James Beattie and was a livewire presence throughout.