The Gunners did not find an opener until the 75th minute when Pierre-Emerick Aubameyang converted from the spot and after the forward soon netted a second, Alexandre Lacazette completed the scoring with a penalty of his own.
Arsenal close the gap on fifth-placed Chelsea to five points and although the scoreline was harsh on Stoke, the Potters remaining three points away from safety.
Stoke went into the game having not won since January 20, but the visitors almost took an early lead in North London as Xherdan Shaqiri sent a curling effort from 20 yards marginally wide of the far post.
Arsenal were giving away possession too cheaply and a number of misplaced passes led to groans from the home support, but the Gunners almost went ahead in the 23rd minute when Aaron Ramsey's close-range volley clipped the top of the crossbar before going over.
However, aside from a speculative volley from Hector Bellerin and two headers being sent off target by Calum Chambers, Arsenal struggled to create anything in the final third as Jack Butland remained without much to do in goal.
Stoke were managing to threaten at the other end but like their hosts, their final ball was lacking and Arsenal went into the break grateful that David Ospina did not have to make a save throughout the opening 45 minutes.
Both teams continued to find if difficult to create openings after the restart and it took until the 57th minute for a chance to fall Arsenal's way, with Danny Welbeck scuffing a volley wide of the post.
Moments later, Badou N'Diaye made an excellent block to deny Mohamed Elneny while Nacho Monreal forced a good save out of Butland from long range, but Stoke almost stunned Arsenal as Shaqiri's inswinging corner struck the far post and rebounded clear.
That near miss sharpened Arsenal's focus with Mesut Ozil setting up Aubameyang - who fired straight at Butland - while Chambers did the same when meeting Monreal's flick on from a corner.
However, Arsenal finally got their goal through Aubameyang, who sent Butland the wrong way from the spot after Bruno Martins Indi was adjudged to have tripped Ozil inside of the penalty area.
From the resulting corner, Arsenal got their second when a cross into the box found Aubameyang around 12 yards from goal, and the Gabon international made no mistake with a clinical finish into the far corner.
The Gunners were soon awarded a second penalty of the match as Lacazette was pushed over by N'Diaye and the Frenchman got back to his feet to slot into the corner on his first appearance since the end of February.
The goal capped a positive afternoon for the home side with Wenger resting a number of his players ahead of the first leg of their Europa League quarter-final with CSKA Moscow on Thursday night.
ARSENAL (4-2-3-1): Ospina; Bellerin, Chambers, Mustafi, Monreal; Elneny (Xhaka 76'), Wilshere (Mkhitaryan 76'); Ramsey, Ozil, Welbeck (Lacazette 61'); Aubameyang
STOKE CITY (4-4-1-1): Butland; Johnson, Shawcross, Martins Indi, Pieters; Bauer, Badou, Allen, Ramadan (Crouch 80'); Shaqiri; Diouf (Berahino 61')