Díaz's father was released hours before the match, 12 days after his kidnapping in northern Colombia by members of the guerrilla group National Liberation Army.
Diaz had appealed for his father's release after scoring Sunday in a Premier League match, revealing a T-shirt saying "Freedom for Papa" in Spanish.
On Thursday, Díaz didn't score but still tested Toulouse goalkeeper Guillaume Restes with a low right-footed shot from outside the area in the first half.
Aron Dønnum scored in the first half and Thijs Dallinga doubled the advantage just before the hour mark for the hosts, who lost 5-1 at Anfield two weeks ago.
An own goal from Cristian Cásseres made it 2-1 in the 74th minute but Frank Magri quickly restored the two-goal lead.
The game was Juergen Klopp's 450th as manager for Liverpool.
"It was well deserved to lose because they won pretty much all the decisive challenges, battles," Klopp said. "We have too many situations where we should have won the ball but we didn't.
"I only saw the video back now and for me it's not a handball, but how can I decide that?
"Actually, I am a bit more concerned about (the fact that) I would have loved us to have played better, to be honest. That's my main issue tonight."
The Reds lead Group E with nine points from four games. Toulouse trail by two in second.
Leverkusen remained the last team with a perfect record after four games with a 1-0 victory at Qarabag in Group H. The Bundesliga leaders reached the round of 16 after Victor Boniface scored the winner from the penalty spot in stoppage time.
Slavia beat Roma 2-0 in Prague, handing José Mourinho's team their first defeat. Václav Jurečka and Lukáš Masopust netted second-half goals for Slavia to tie Roma atop Group G with nine points each.
At the Amsterdam Arena, Brighton took control of their Europa League destiny with a commanding 2-0 win against Ajax to complete back-to-back victories over the Dutch giants.
It was a performance that matched the ease with which they dispatched the four-time European champions at home two weeks earlier.
Since then, Ajax had appointed a new manager in John van't Schip, but despite an uptick in domestic form they never looked like they had the firepower to unduly trouble Brighton, who could have won by a greater margin had Joao Pedro, Adingra and Fati not wasted good chances.
David Moyes' Hammers top Group A with Freiburg, 5-0 winners over Backa Topola, on nine points.