Late goals from Jefferson Montero and Ki Sung-yueng almost guaranteed that Swansea will end the season in the top eight – their highest-ever finish in the Barclays Premier League.
But Swansea are now only four points adrift of Southampton following their surprise defeat at Sunderland – and seventh place could be enough for Europa League qualification if Arsenal beat Aston Villa in the FA Cup final at the end of May.
“The carrot (of Europe) is there for us as we have got Arsenal away, Man City home and Crystal Palace away. They are tough games,” Swansea manager Monk said.
“We are under no illusions how tough it will be, but if we perform like we did in the second half, in those games we will be a handful for anyone.
“It is feasible to get into Europe but we keep our feet firmly on the ground here.
“This game was about consolidating our place in the top 10 and we are in control of that eighth spot now.
“We have beaten our best points tally and you have to find new things to fight for.”
Ecuador winger Montero broke the deadlock with a first goal in English football since his £4million summer switch from Mexican club Monarcas Morelia, a 76th-minute diving header at the far post.
Ki then settled matters with his eighth goal of the season in stoppage time after coming off the bench in the closing stages.
“I don’t think Jeff has scored too many goals with his head and everyone is happy for him,” Monk said.
“I had a few words with them at half-time as the first half we were at 70 per cent.
“The tempo of our passing was not where it should have been but we were so dominant in the second half and deserved to win the game.”
Stoke had defender Marc Wilson sent off in the 85th minute for two bookable offenses but manager Mark Hughes felt his side paid the price for lacking an attacking edge.
“The second half we didn’t really ask too many questions of Swansea,” the Welshman said.
“I thought we were very comfortable in the first half. We had the majority of possession without really having that cutting edge in the final third.
“Our retention of the ball wasn’t quite good enough to get people up the pitch and in the box.
“Our creative play was missing and that was disappointing because I felt it was a real opportunity to get something out of the game.”
On Wilson’s dismissal – the first time Stoke have had a player sent off in the Premier League this term – Hughes said: “The first was a coming together and probably looked more than it was.
“It was a foul but I’m not sure if it was yellow card. But when you’ve got a yellow card you’ve got to steer clear of those situations Marc found himself in.
“He’s tripped the lad going through and unfortunately he’s had to go off, and from that point it was difficult to affect the game.”