The Royals have not won at home in the Championship since January 27 when leaders Wolves were beaten 1-0 and they could not end that run on Saturday as Barnsley grabbed a stalemate.
Mick McCarthy's side have gone on to clinch promotion but Reading fear they could be pushed out of the play-off positions by any two from Burnley, Swansea and Preston.
Coppell said: "We have got to get at least three points from our remaining three games. That is something we are capable of doing. If we don't then we are not good enough and we will end up staying where we are."
The draw was also Reading's seventh successive home game without a win.
Coppell said: "It's massively disappointing once again. It's the strangest run of results at home I have known at any club I've been at.
"It's been the strangest season in many ways. Earlier on when we were winning games I was thinking 'how on earth were we winning those by threes, fours and fives?'.
"We were poor in the first half. It was almost as though there was nothing at stake for either team. It was shadow boxing.
"In the second we were what I expected us to be and we had opportunities. We just didn't have our shooting boots on."
Dave Kitson and Jay Tabb saw efforts saved by Reds goalkeeper Luke Steele, who was picked for the first time since September.
Indeed the only time Steele was beaten was just before the hour mark when Barnsley defender Rob Kozluk sent a back-header over him and was grateful to see it bounce beyond the back post.
Barnsley threatened only occasionally although Anderson de Silva was only just wide with an effort from the edge of the box midway through the second half.
The point kept Barnsley from dropping into the bottom three as although Nottingham Forest climbed above them by beating Coventry they are now ahead of Norwich, who play tomorrow.
Barnsley manager Simon Davey backed his players to beat the drop.
He said: "We know what we have to do and we have three games to do it. We know what points tally we need and this is another point towards it.
"This is a confidence-booster. The performance was good, the resilience was good and everybody was mucking in and working hard.
"I always have been confident of staying up, from day one of pre-season, and my players are exactly the same."
Davey had opted to drop first-choice goalkeeper Heinz Muller to give Steele a rare outing.
He said: "We have two fantastic keepers here and they have been vying for that number one shirt all season. Luke has been champing at the bit to play and stepped up to the plate."