The Toffees were thumped 3-0 by Atalanta in their first Group E game and never impressed against a resolute Apollon side.
Adrian Sardinero took advantage of some poor defending from Ashley Williams to give the visitors a deserved lead in the 12th minute before Wayne Rooney equalised after an equally shambolic piece of play from Yuste.
Nikola Vlasic, who replaced Idrissa Gueye at the break, then scored his first goal for the club midway through the second half, but, after having Valentin Roberge sent off late on, Apollon secured a point through Yuste's 88th-minute goal.
Ronald Koeman's men were made to pay for a sluggish start after just 12 minutes as the visitors took full advantage of some sloppy defending from Williams.
The former Swansea City defender's clearance was pounced on and played out to Anton Maglica, whose teasing cross from the right was met by Sardinero. His initial shot squirmed through Jordan Pickford's attempted block and the striker was able to tap in from almost on the line.
Apollon very nearly scored a second minutes later, but Giorgos Vasilou's spectacular half-volley from distance arrowed narrowly past Pickford's far post. Rooney then pulled the hosts level in the 21st minute and he is unlikely to have scored an easier goal in his illustrious career.
Under pressure on the right, Yuste elected to pass back to his goalkeeper. He had horribly misjudged Bruno Vale's position, though, and only succeeded in teeing up Rooney to slot home into an empty net from all of six yards.
The goal appeared to spark Everton into life and they came close to a second soon after through Gylfi Sigurdsson's volley, which was well held by Vale, and a curling effort from outside the area by Rooney.
Vale was twice called into action again just before the break as he first theatrically pushed away Tom Davies' dipping shot from distance, and then punched clear Sigurdsson's free-kick from the edge of the area.
Koeman brought on Vlasic at half-time in a bid to inject some pace and purpose to his side's attack, but it was the visitors who threatened first as Antonio Jakolis' long-range drive was repelled by Pickford.
1 - Everton are the first English side to pick up just one point after their first two Europa League group stage games. Glum. pic.twitter.com/gEXEi5LLRC— OptaJoe (@OptaJoe) September 28, 2017
Vlasic did then repay his manager's faith in the 66th minute as he latched onto Sigurdsson's throughball to coolly slot past the advancing Vale.
It marked Sigurdsson's first assist for the club since his big-money move from Swansea.
Roberge received his marching orders with just five minutes remaining for a stamp on Dominic Calvert-Lewin, but there was a late sting in the tail for Everton.
Joao Pedro's inviting free-kick was duly met by Yuste to send the 1,500 travelling fans into delirium. There was still time for Calvert-Lewin to squander a golden opportunity to grab a winner when he headed straight at Vale from six yards out, but it would have been unfair on Apollon, who were good value for their point.
Key Opta stats:
- After losing each of their last five Europa League group stage matches away from home, Apollon Limassol have picked up a point for the first time on the road since October 2013.
- In fact, the Cypriots are unbeaten in their first two European group stage games for the first time in their history (D2).
- There were just 118 seconds between Apollon Limassol's red card and their equaliser against Everton.
- Wayne Rooney is the first player to score a Europa League goal in consecutive seasons, but with different English clubs since Clint Dempsey (Fulham 2011-12, Tottenham 2012-13).