Second-half goals from Mohamed Salah and Sadio Mane earned the Reds the Champions League last-16 victory in Budapest, ahead of the return leg in Liverpool on 11 March (AEDT).
However, prior to those goals Alisson came up with a big save in the 46th minute to deny Christopher Nkunku.
The clean sheet comes after the Brazilian goalkeeper was criticised for his turnovers in the 4-1 loss to Manchester City and a defensive mix-up in the 3-1 defeat to Leicester City in the Premier League.
"Definitely, definitely," Klopp said when asked about Alisson feeling better. "He honestly had big moments in the Leicester game also, some brilliant saves. So that's how it is, but of course it helps.
"A clean sheet, I can imagine Ali will sleep really well on the plane on the way home, and as always he helped us a lot."
Liverpool’s triumph snapped a run of three consecutive defeats amid a period of tumult for Klopp's side as their Premier League title challenge has fallen apart, slipping down to sixth.
The German manager was delighted with his players' response against last season's Champions League semi-finalists, which provided a morale boost although he insisted that was not a tonic for their problems.
"Look, we are not children anymore so we would now think 'OK, now we won that game and everything is fine again'," Klopp said.
"For me it is much more important that we played again good football. Again, not the first time, in the last few weeks, I understand that and I respect that 100 per cent, that nobody is really happy with the results, that's clear."
Klopp's caution is understandable given last month's form dip was halted with wins over Tottenham and West Ham before their latest slump.
He said: "I know Leipzig pretty well and I know how dominant they can be, football-wise, and how overwhelming they can be in a game from a physical point of view.
"That is what we had to deal with tonight and the boys played a really good game, a really good game."