Carayol, a 27-year-old winger, made his debut for Leeds after joining from Championship rivals Middlesbrough on loan until the end of the season.
Coyle, who is only 20 years of age and plays as a right-back, started the FA Cup third-round tie against Rotherham and played for the entire 90 minutes.
The duo played well and helped Leeds win the tie 2-0 at Elland Road.
Carayol put the Whites in the lead on the stroke of half time when he lashed a dipping shot past Lee Camp from the left edge of the box.
Souleymane Doukara added the second for Steve Evans’s side in the final moments of the game.
Leeds fans have taken to Twitter to have their say on the performances of Carayol and Coyle against Rotherham.
Below are some of the best comments:
@WYSdaily Mustapha Carayol had a great debut. A superb goal by him. My Man of the Match is Carayol. I want Man Utd in the 4th round draw— Lily Oloumi (@LilyOloumi) January 9, 2016
Well deserved win for #LUFC. Hard to beat and competing across the field. Great debuts today for Coyle and Carayol. A good day— Mark Wilson (@MarkWilsonRadio) January 9, 2016
Not the best of games to say the least but a wins a win. Fantastic debut goal from Carayol and Douks on the scoresheet. Can't complain.— LUFC Talk (@Talk_lufc) January 9, 2016
Scrappy performance, carayol looked positive. Coyle had a strong performance too. Better draw for next round please #lufc— Isaac (@isaackeighley) January 9, 2016
The Carayol Evans link working as intended wonder what all the knockers / doubters have to say now? #LUFC— For the fans (@St_Jigzy) January 9, 2016
Better second half, great strike from Carayol, impressed again with Coyle #lufc— Future Whites (@futurewhites) January 9, 2016
Wasn't pretty today but a win is a win and its now 8 unbeaten. Lewis Coyle was great another great product of the academy #lufc— Sean Bugg (@Silver_Bugg) January 9, 2016
Coyle did what I expected shows he's our future RB just how Taylor stepped in he's going to have a superb furter #LUFC— For the fans (@St_Jigzy) January 9, 2016