Robbie Brady urged the Clarets' fans to keep the faith after they witnessed another away trip without reward.
The travelling support was again out in full force at Goodison Park as Burnley looked to stop the Everton bandwagon.
Those fans were repaid with a positive performance from Sean Dyche's men before a quick-fire double mid-way through the second half secured the Toffees' eighth successive home league win - a Premier League club record which leaves the Clarets with just two more opportunities to gain a first away win.
"They have been excellent and followed the team everywhere. Since I've been here it's been amazing to watch," said Brady.
"Even today it was packed to the rafters over in that corner. I say just stick with us. We'll grind out these results out and we'll be okay.
"We felt we could have taken something from the game. We didn't manage it in the end but there are plenty more opportunities to come and we'll relish it.
"It's a tough one to take. In the first half we put them under pressure and created some good chances.
"It's disappointing in the end because I thought we put in a decent performance. We're not too far off it.
"It's always difficult to come here anyway but when they're on such a good run as well, to perform like we did today is a credit.
"But they are seven games unbeaten for a reason and if we give them half-chances like that they have more than enough to put them away."
Burnley were the better side before the break and even after going behind to a Phil Jagielka header just after half-time, hit back through a Sam Vokes' penalty.
But a Ross Barkley shot, cruelly deflected in off Ben Mee, and a top-class finish from the predatory Romelu Lukaku nipped the fightback in the bud.
"We were disappointed to go 1-0 down but we showed good character to go back down the other end and win the penalty and equalise," added Brady, whose growing influence was apparent as he helped take the fight to the Toffees.
"We felt we were right in the game and it was a bit more like us today and it felt good all-round.
"There were just a couple of little details and then we found ourselves 2-1 behind and couldn't get ourselves re-grouped quickly enough and Lukaku managed to put a third one in.
"When you concede two goals so close together it makes it difficult, although even after that we created a couple of chances.
"We were disappointed in the end but we can definitely take some positives from it."