Roberto Martinez's side were good value throughout the game, but couldn't take their chances when they arrived as they felt the absence of Romelu Lukaku and Lacina Traore.
Unsurprisingly under Martinez, Everton had more possession than their hosts, but failed to make it count and leave West London empty handed.
A late free kick conceded by Phil Jagielka was turned home, though there is some confusion over whether the goal will be credited to Terry or as an own goal by Everton goalkeeper Tim Howard.
Everton fans were understandably gutted to concede such a late goal, and Martinez, midfielder Ross Barkley and winger Kevin Mirallas came in for criticism from the supporters, though Sylvain Distin, Gareth Barry and James McCarthy were singled out for praise.
Here are some of the post-match reactions from Toffees fans.
Am I exaggerating in saying I think we'd have hung on without Barkley in those final 10minutes? #efc— Joanna Aicher (@JoannaAicher) February 22, 2014
if you cant score goals you dont win games, dnt matter how well you play.. #efc— Simon (@JA_EFC) February 22, 2014
Exactly same as Spurs and come away with nothing. Lack cutting edge but played brilliantly at times. Unlucky #efc— steve callaghan (@steve_cally) February 22, 2014
John Terry ..a more prolific striker than Steven Naismith Martinez bringing great things to this club but letting Jelavic go was insane #efc— Paul J Hey (@pauljhey21) February 22, 2014
This is what happens when you don't have a striker. Two games on the trot we've dominated and nothing to show for it #efc— Daniel Button (@D_Button) February 22, 2014
Result aside,Gareth Barry and sylvan distin have been absolutely brilliant today! They all have to be fair, but them 2 the best for me! #EFC— Aaron Mcconnell (@aaronmc26) February 22, 2014
Mirallas should be taking his opportunity upfront in games like this. Instead he flatters to deceive again. Need a much more from him #EFC— Danny Mullally (@DanielMullally1) February 22, 2014
James McCarthy is a special footballer. Had my doubts but he's been a class act all season #efc— S. (@OGrianain22) February 22, 2014
Gutted. Didn't deserve to lose that. Bold subs by Martinez, but the game was there to be won. #EFC— Chris McCann (@chrismccann72) February 22, 2014
Great performance, but overall that's top 4 gone, and In all honestly I don't want Europa league! #EFC— DRaynard01 (@DannyRaynard) February 22, 2014