Middlesbrough had more than their fair share of the play but once again looked lightweight in attack.
Turkey international Tuncay strove hard but once again lacked support. Dong-Gook Lee is not the answer to Southgate's attacking problems and the Boro boss will no doubt be trying desperately hard to strengthen his strikeforce in the transfer window.
Not that Southgate offered much encouragement in his programme notes as he wrote: "January is a notoriously difficult time to buy. If it proves possible, and the right quality of player becomes available, it will be a huge lift and bonus for everyone.
"But we have to work on the basis that we have got the squad that we have got. There is a great challenge for everyone to finish in a higher position than we are in at the moment.
"There is no benefit to anyone being negative. We are in a situation we don't like, but we were in an even worse situation three weeks' ago."
It was inevitable that Everton striker Yakubu would receive a hostile reception from the home fans following his acrimonious departure from the Riverside Stadium last summer.
The £11.25million striker had a subdued game and never looked like adding to his impressive goal tally this season.
The Nigeria international suffered ironic cheers from home fans when he was withdrawn in the closing minutes of the game.
Everton midfielder Lee Carsley was soon in the referee's bad books when he was cautioned after only three minutes for a foul.
Middlesbrough, who were hoping to follow up their 1-0 win over Portsmouth on Saturday with another three-point haul, combined well but lacked a cutting edge.
Quick tackling by defences curbed the attacking threat of both teams and neither goalkeeper had much to do in the opening stages.
Middlesbrough's first serious shot of the game came after 30 minutes when Stewart Downing's effort struck Phil Jagielka on the goal-line.
Everton snatched the lead in the 67th minute when Andrew Johnson held off the challenge of two defenders before firing into the far corner of the net.
The visitors wrapped up the three points with a second goal five minutes later following Downing's mistake. James McFadden accepted a return pass from Johnson before shooting home.
Everton fans rubbed salt into the wounds of the home team in the last few minutes as they chanted: "Going down, going down".