The Swans headed into the encounter at the Liberty Stadium on the back of a 1-0 victory over Arsenal away from home at the Emirates Stadium in North London.
There was confidence among the Swansea players of getting the better of City, but in the end they failed to pick up the three points.
It proved to be a hugely entertaining and thrilling game of football, but it was Manuel Pellegrini’s side who were celebrating at the end, having won it 4-2.
Following the defeat, Gomis - who scored for the Swans in the 64th minute - and Barrow posted the following messages on Twitter:
4 this last home game we've lost against a great team of Manchester City Despite this I want to thank our wonderful fans who helped us a lot— Bafétimbi Gomis (@BafGomis) May 17, 2015
I 'd like to thank them for welcoming me and allowing me to feel good That helped me a lot to score some goals in this second part of season— Bafétimbi Gomis (@BafGomis) May 17, 2015
Thank you for everything! I'm jack more than ever!!!— Bafétimbi Gomis (@BafGomis) May 17, 2015
YEEESSS!!! GREAT win over a good side. Proud of this team!! Wouldn't have been easier without #JackArmy, thanks for the massive support!— Modou Barrow (@modoubarrow14) May 11, 2015
Not the result we wanted but a huge thanks goes to the #JackArmy for their massive support this season! It's been a 1st good season for me.— Modou Barrow (@modoubarrow14) May 17, 2015
The season has been brilliant with the support from my teammates & the entire Club staff. Next season will be much better hopefully. Tnx!— Modou Barrow (@modoubarrow14) May 17, 2015
Swansea are now eighth in the Premier League table with 56 points from 37 matches, four points behind Southampton, who are seventh in the standings at the moment.
Garry Monk’s side will face Crystal Palace away from home at Selhurst Park in their final game of the season.