Goals from Romelu Lukaku and Barkley within the space of five minutes put the Toffees back in the driving seat, and they managed to hold on for all three points despite a late Ashley Williams header giving Swansea hope.
Here, Sports Mole takes an in-depth look at an absorbing encounter on Merseyside.
On target: 6
On target: 7
Was the result fair?
A quick look at the stats will tell you that Everton were fortunate to get away with all three points from this match. In truth, a draw probably would have been the fairest result, but Swansea have only themselves to blame for not coming away with anything. The visitors created plenty of goalscoring chances but were nowhere near clinical enough with them. Tim Howard did have to make a couple of smart saves but, on the whole, Swansea's poor finishing was the biggest reason behind the scoreline.
It was certainly not the most free-flowing and fluid Everton performance that you will see under Roberto Martinez. They usually like to dominate possession but, playing against a side like Swansea, that was always going to be difficult. They didn't help themselves in that respect, however, giving the ball away far too cheaply on numerous occasions. Defensively they were very shaky at times and Martinez will know that teams with more of a cutting edge would have punished them this afternoon. All in all, the hosts can improve greatly on their performance, but the fact that they were not at the best yet still managed to score three goals and win will be pleasing for the manager.
For the first half an hour or so, Swansea were second best to Everton. However, as soon as Bony scored their equaliser they began to take control of the match, and Everton were happy to hang on until the break still level. Swansea also started the second half well, until they were hit by two goals in the space of five minutes that put Everton in control. They soon found their feet in the match and played some nice football but, as has often been the case this season, they were wasteful in front of goal. They carved out a whole host of chances as the match wore on, and Garry Monk will be wondering just how his side did not manage to convert one more of them to at least nick a draw.
Sports Mole's man of the match
Ross Barkley: In truth, Ross Barkley didn't have his best game today, misplacing a number of passes in midfield and particularly struggling at times when Swansea were on top. However, he picks up this award for his contribution to both the opening goal and his winning goal. He produced a lovely bit of footwork to draw the foul from Chico Flores and win the penalty, which Baines duly dispatched, while he did well to lose the defender again to nod in from close range for Everton's third.
This one goes to Chico for his foul on Barkley for the penalty. Barkley threw a few step-overs before knocking the ball away from the defender, who lunged at the thin air and only succeeded in bringing the youngster down. It was a nice piece of skill from Barkley, but a defender of Chico's experience should not be making such a rash challenge inside his own box. He could also win this dubious honour for his part in Everton's third, where he lost Barkley, allowing the midfielder to nod home from close range.
Michael Oliver had a good game today and kept his cards in his pocket throughout, not producing a single yellow. The biggest decision was, of course, the penalty, and he got what initially looked to be a tough decision spot on. There was no doubt that it was a foul by Chico, but it is unclear on first viewing whether it was inside the box or not, although replays showed that Oliver was right to award the spot kick.
Everton: Next up for the Toffees is a trip to face Newcastle on Tuesday.
Swansea: Things don't get any easier for Swansea, meanwhile, as they travel to London to take on Arsenal, also on Tuesday.