The fifth-placed Gunners went into the contest at a ground where they had won just once in their previous eight Premier League visits knowing only a victory would really do in their quest for Champions League football.
They appeared comfortably on course for three points after Mesut Ozil put them 2-0 up in the 55th minute, adding to Olivier Giroud's first-half opener.
But Stoke then applied some pressure and pulled a goal back through substitute Peter Crouch's controversial 67th-minute effort that saw the ball go in off his hand.
Arsene Wenger's men responded to that setback in fine style, keeping the Potters at bay before strikes from Alexis Sanchez and Giroud wrapped up the win.
And Arsenal full-back Bellerin, who provided assists for his side's first and third goals, told the club's official website: "That's something this team has.
"We have a lot of heart, a lot of character and, even though in some games it may not look like we've shown it, the team really wants to be there (in the Champions League) and we've shown that (against Stoke).
"We knew that we were playing for our lives really, so any result other than a win wasn't good for us.
"We knew that at the start of the game, so we knew we had to win no matter what
I think the team did really well."
Gunners defender Rob Holding echoed Bellerin's sentiments in terms of how the team handled things after Crouch scored.
"He got that goal and their fans got a bit louder, so we showed a bit of character to stay in the game when they had seven corners in a row," Holding said.
"We just had to withstand it and then score more goals on the break to polish it off."
On Crouch's goal, he added: "I didn't see it go in, but I saw some of the lads shouting and waving their arms about
It went in but we got the result, so it doesn't matter."
Crouch admitted the ball had struck his hand when he scored, but was only sorry Stoke - victorious in just one of their last 10 games - could not salvage anything from what was their final home match of the season.
"I just threw everything at it and it's come off my hand a little bit but it was a goal and I was pleased to score another one," the veteran striker told Stoke City Player regarding his ninth goal this term.
"It's just a shame we didn't get the result we would have liked.
"I thought we were going to get back into the game but with us throwing men forward you know that Arsenal can pick you off and that's what happened in the end.
"I don't think we've been at it towards the end of the season and it's been very frustrating, certainly to watch and be a part of.
"It's a shame we didn't finish as well as we would have liked in front of our own fans but we'll go again next season.
"We will regroup, look at a few things and be a lot stronger next year."
Source : PA