Tigers' third game in eight days following defeat in the EDF Energy Cup final against Ospreys and a loss at Bath in the Guinness Premiership brought a 32-14 win that was effectively secured by three tries inside the opening half hour.
Leicester's bonus point win took them above London Wasps and Shale Sharks.
That could all change again on Sunday with Wasps travelling to Saracens and Sale entertaining Newcastle Falcons.
Leicester's last game is at home to Harlequins on May 10 and that could prove to be crucial with Dean Richards' team slipping up against London Irish.
Defending champions Leicester had claimed just 12 points from their previous seven games and Loffreda said: "This was a relief.
"We had two defeats in the last week and everybody was anxious.
"It was very important to the club and the team to put in a performance like this.
"We have finished a tough week with a good win and five points.
"We didn't want to have a schedule like this, but we have had to deal with it.
"But we will have to play better than that in our last two games against Newcastle and Harlequins if we want to be competitive."
Loffreda added: "The players played with a lot of commitment. They worked very hard.
"But this is just the first step and there are two more steps to go."
Bristol assistant coach Paul Hull was pleased with his young side's performance.
They rested several players and Hull was particularly pleased with the way his side responded in the second half after trailing 20-0 at the interval.
They grabbed two second-half tries - Shaun Perry crossing for the first to strike a blow in his duel with Leicester number nine Harry Ellis - and Hull said: "We came with a young side and they showed some true grit and team spirit in the second half.
"There were some harsh words at half time and the boys responded to win the second half 14-12.
"We are disappointed. We didn't come here to lose.
"I was pleased with some of the young players. It was a good step towards next year.
"They got some game time in front of a big crowd."