Connor's side are eight points adrift of safety with just four games remaining, and their next fixture pits them against title-chasing Manchester City.
Their prospects of avoiding relegation to the Championship therefore appear bleak, but Connor will not be throwing in the towel until survival is mathematically impossible.
The former Wolves assistant is still to claim his first win since replacing Mick McCarthy, yet he claims there were positives to be drawn from his side's spirited showing at the Stadium of Light on Saturday.
"We'll keep going and keep fighting to the very end," Connor said.
"I asked the players to make sure we were competitive in all 38 games, so we'll try to keep picking up points to make sure all 38 games mean something at the end of the season.
"The pressure's not off because we can still stay up. It's not impossible.
"Looking at it, to have all 38 games competitive is what we want and we want to get to Wigan (on the final day of the season) still with some sort of chance of staying in the league."
Wolves were the better side against Sunderland and would have claimed a rare victory had Simon Mignolet not made an excellent second-half save to keep out Steven Fletcher's 81st-minute header.
Wayne Hennessey was able to celebrate a first Premier League clean sheet since the third game of the season, and despite their position at the foot of the table, there appeared little wrong with the spirit of the Wolves squad.
Connor said: "The players are an honest bunch and they've been brought up that way. They're hard working, but of course it's difficult with where we are in the league.
"Performances have been lost because of the results, but they've kept going.
"We've made errors at the wrong times which have cost us, but this was a good performance and we could have nicked it with a goal at the end."
Sunderland's best opportunity saw James McClean denied by a sharp low save from Hennessey, and the hosts lacked the spark they have displayed in the majority of their matches under Martin O'Neill.
The Black Cats have hauled themselves clear of the relegation zone since the Northern Irishman replaced Steve Bruce in December, but they remain a work in progress and significant changes are anticipated in the summer.
"We want to try and progress," O'Neill said. "We've pulled ourselves round from potential relegation, but we have work to do.
"We want to try and improve on what we've done and that would be our ambition.
"It might be in the last couple of games that we might be able to start other players in the side and see how that goes to have a look at them to have a viewpoint for the summer, but we would need to strengthen."
Source: AFP
Source: AFP