Madrid's win should end talk of crisis following a poor start to the club's domestic season.
Goals by Gareth Bale, Cristiano Ronaldo, James Rodriguez, Karim Benzema and a Basel own-goal secured the 10-time champions' a very comfortable victory in Group B.
Five years after its last involvement in the Champions League, Liverpool needed a penalty from Steven Gerrard in the third minute of added time - just two minutes after Ludogorets looked to have snatched a draw - to earn all three points.
Elsewhere, Borussia Dortmund beat Arsenal 2-0 and last season's finalist Atletico Madrid lost 3-2 at Olympiakos.
Madrid's win was much easier as Carlo Ancelotti's players silenced their critics with a beautiful display of free-flowing football, turning the page on a poor run of form.
Madrid lost to Atletico in the Spanish Super Cup last month before Ancelotti's side collapsed 4-2 at Real Sociedad in the league and followed up with a 2-1 loss to Atletico at home on Saturday.
"I liked the team's reaction, I think the Bernabeu was happy," Ancelotti said.
Against Basel, Madrid played with pace and was spectacular up front, despite the defence looking anything but solid.
Madrid opened the scoring in the 14th minute when Jose Fernandez's shot was deflected in by Marek Suchy for an own-goal.
Bale then received a clever ball from Luka Modric to double his team's lead and provided Ronaldo with a pass which the Portugal forward — the competition's top scorer last season with 17 goals — converted a minute later.
Rodriguez added the fourth in the 37th before Derlis Gonzalez pulled one back for Basel in the 38th. Benzema made it five in the 79th after a pass from Ronaldo.
Mario Balotelli got his first goal for Liverpool in the 82nd minute before Ludogorets, a side who had not won a game in UEFA competition before last season, equalized through Dani Abalo.
Liverpool was then awarded a penalty when goalkeeper Milan Borjan brought down Javier Manquillo and Gerrard slotted home from the spot.
"Looking at the game as a whole, we did OK. We didn't do better than OK," Gerrard said. "I thought they caused us quite a few problems on the counterattack, so there's a lot for myself and the lads to learn."
In Group A, former Manchester United striker Carlos Tevez ended his European goal drought as he scored twice in a 2-0 win over Malmo, for his first goal in the competition since April 2009.
The group's other game provided a shock in Athens, where Olympiakos defeated Atletico after Arthur Masuaku — with a superb strike from outside the area — and Ibrahim Afellay scored early goals.
Mario Mandzukic pulled one back before Kostas Mitroglou extended the host's lead. Antoine Griezmann's late strike from close range for Atletico did not prevent Olympiakos from earning the three points.
Monaco won against the run of the play with a 1-0 victory over Leverkusen in Group C courtesy of Joao Moutinho' goal, while Benfica's 51-match unbeaten home record in all competition came to an end.
Hulk and Axel Witsel scored first-half goals to give Zenit a 2-0 win against 10-man Benfica after goalkeeper Artur was sent off in the 18th for obstructing Danny.
In Group D, Borussia Dortmund strikers Ciro Immobile and Pierre-Emerick Aubameyang scored either side of halftime to give their side victory over Arsenal.
Immobile, signed from Torino in the off-season, sprinted almost half the length of the pitch and held off Arsenal defender Laurent Koscielny before dispatching a low shot inside the far post just before the break.
"I'm very happy to score, that we won, because it hasn't been easy for me to settle into the Bundesliga. I'm happy that I was able to make my teammates happy," Immobile said.
Galatasaray striker Burak Yilmaz scored in stoppage time to snatch a 1-1 draw against Anderlecht in the group's other match.