"We have to finish in the top four, otherwise we don't think we've been successful," Barker said after his team's win in Pretoria.
Goals in either half from Sibusiso Vilakazi and Papy Faty handed the Clever Boys the hard-fought victory, as they climbed to fourth spot in the league standings.
Top-flight newcomers AmaTuks were left in ninth place with two matches remaining in the campaign.
The major talking point before the start of the game was the inclusion of 45-year-old veteran goalkeeper Andre Arendse in the Clever Boys' starting line-up, with first-choice shot stopper Emile Baron unavailable due to injury.
Baron's injury added to their deepening goalkeeping crisis, with Ryan Harrison (fractured finger), Jackson Mabokgwane (knee injury) and Steven Hoffman (broken ankle) all sidelined, prompting the Students to register Arendse as a squad member, in order to serve as cover.
"It was always going to be tough when you have a goalkeeper coming out of retirement and who hasn't played for a couple of years," Barker said.
"I thought the pressure on a young goalkeeper would be too much tonight and that's why we convinced Andre to play in this game.
"Whether he'll play in the next game, it's going to be up to him, but we're hoping that one of the two younger keepers will come on.
"But he gave us that protection tonight and I think he did a marvellous job."
Having squared off against his nephew, Tuks coach Steve Barker, Clive Barker heaped praise on his late brother's son.
"We love Stevie and he's such a credit to his family," Clive Barker said.
"His dad was my brother and he loved him (Steve) very much, and so did his mother, and they've both passed away unfortunately.
"I think he's a very, very good coach and a better person and we're lucky to have him around."
Clive Barker boldly predicted Wits could see themselves in the running for league honours next season.
"It's nice that the results have gone the right way," he said.
"If we can get over this goalkeeper setback, then I have no doubt that Wits, if they buy sensibly, will win the league next year."