The Spaniard bagged a 52nd minute equalizer to cancel out Peter Crouch's first-half opener on a blustery night at the Britannia Stadium.
Chico Flores scored a 52nd-minute leveler after Peter Crouch had given the home side the lead in the first half.
The result sees the visiting team move into the top half of the table, leapfrogging Aston Villa and West Ham on goal difference, while Stoke sits three points above the relegation zone.
Mark Hughes made two changes to the side that drew at Southampton on Saturday, while Pablo Hernandez was rewarded for his performance from the bench in Swansea's 3-0 victory over Cardiff City with a place in the starting lineup.
That derby triumph was Monk's first game at the helm and his side began brightly in a delayed kickoff at a windy Britannia Stadium, with traffic problems having caused the game to be moved back 15 minutes.
Nathan Dyer's teasing cross from the right gave the home side an early scare, looping over Asmir Begovic and onto the roof of the net.
But it was Stoke, enjoying a run of just one defeat from its previous nine home league games, which took the lead with 17 minutes gone.
Peter Odemwingie was played in by Crouch, only for the former West Brom striker's effort to crash off the post and into the path of the former England forward, who rifled a deflected shot through a crowded six-yard box and into the net.
Swansea should have been level six minutes later when Wayne Routledge's whipped cross fell to Jose Canas at the back post, but the Spaniard could only blaze it over the bar.
Dyer continued to cause problems down the right side, and was handed another chance seven minutes before halftime when he cut inside to send a low drive wide of the post.
Wilfried Bony came into the match having not scored a Premier League goal on the road all season, but was handed a golden opportunity just after the break.
A neat one-two with Routledge gave the Ivorian some space on the edge of the box, but he sent his effort high and wide.
Parity was restored in the 52nd minute when Hernandez picked out Chico inside the box and he rose highest to flick a header past a despairing Begovic.
It was no less than the visitors deserved, but Stoke should have regained the lead just after the hour mark when a teasing ball from Charlie Adam somehow eluded both Steven N'Zonzi and Ryan Shawcross in the six-yard box.
Shawcross made amends 15 minutes later with a superb goal line clearance to deny Bony, and substitute Jonathan de Guzman then forced Begovic into a great save with a well-struck effort from distance late on, but Hughes' side held firm to earn a point.