A late Chris Budgen try swung a tight match in Saints' favour before a Bruce Reihana penalty in the last minute made sure of the win.
Director of rugby Pountney said: "Leeds did not get anything from Gloucester and our four points here mean we are virtually a game in front of them.
"But we know how much pressure there is around us and I am sure that Leeds do as well.
"We are technically nowhere near where we want to be and our long injury list does not help.
"Clearly we have still got a lot of work to do before we play Worcester next week and none of the players are under any illusions about what has to be done."
Bristol had led three times before Budgen's touchdown finally put Northampton in sight of a much-needed win.
Jon Clarke had put Saints ahead with an early try but Bristol hit back with two touchdowns from Samoan David Lemi.
Three Reihana penalties restored Northampton's lead but the visitors got back in front when Darren Crompton crossed.
Carlos Spencer levelled with a drop goal but a Jason Strange penalty looked like winning it until Budgen intervened.
Bristol head coach Richard Hill refused to be too critical of his young side.
He said: "We threw the game away as we had enough chances to win it but it was a lack of composure under pressure that was uncharacteristic.
"Many of this side have not played here before in front of a big crowd and we just did some silly things."
Newcastle, who decided to delay Jonny Wilkinson's comeback following groin surgery for another week, held off a Worcester fightback to win 21-15 at Kingston Park.
Matt Burke scored 16 points and Dave Walder also touched down but tries from Aisea Havili and Tony Windo kept the Warriors in contention.
"We kept nudging ahead and then kept giving them something back and that is why it was so tense in the last 20 minutes," said Newcastle director of rugby Rob Andrew.
"Both teams were just hanging on in there at the end and it was always going to be like that, with both sides having played two games in a week. Both sets of players looked very tired.
"It was a very important win for us. We have now won three in a row in the league and we competed well up front against Worcester, who are not an easy side to play against."