A 4-0 defeat at Chelsea in November saw Allegri face scrutiny merely months after he returned to Turin but a hard-fought victory on Sunday saw them move up to fifth in Serie A.
Cuadrado curled home directly from a corner in the ninth minute and Paulo Dybala wrapped up the points in the 82nd minute with a composed finish.
Juventus remain seven points off Champions League qualification, which is also the case for Fiorentina despite their 3-2 success at Bologna.
Dusan Vlahovic continued his fine goalscoring form with a 67th-minute penalty that turned out to be the winner.
Youssef Maleh and Cristiano Biraghi were also on target for Fiorentina to cancel out Musa Barrow’s first-half effort for the hosts while Scot Aaron Hickey grabbed what proved to be a consolation late on.
Eighth-placed Lazio remain firmly in the race for European football after Ciro Immobile’s brace helped them to a 3-1 win over Sampdoria.
The away side finished with 10 men after Sergej Milinkovic-Savic who had netted in the first half, was sent off with 23 minutes left and former Southampton forward Manolo Gabbiadini gave the home faithful something to shout about with a goal at the death.
Verona produced a remarkable comeback to edge out Venezia in a seven-goal thriller that saw Giovanni Simeone, the son of Atletico boss Diego, end as the hero for the visitors.
Pietro Ceccaroni, Domen Crnigoj and Thomas Henry put hosts Venezia three ahead at the break but Henry put through his own net before Ceccaroni was shown a red card.
Gianluca Caprari reduced the deficit further before Simeone netted twice to complete an excellent turnaround for Verona.
Elsewhere, Spezia and Sassuolo played out a 2-2 draw and both remain in the bottom half of the table.
Freiburg ended their three-game losing run in the Bundesliga and went goal crazy in a 6-0 thrashing of Borussia Monchengladbach.
All six efforts occurred in the opening 37 minutes with Maximilian Eggestein, Kevin Schade, Philipp Lienhart, Nicolas Hofler, Lucas Holer and Nico Schlotterbeck on target.
Stevan Jovetic, previously of Manchester City, hit a double to help Hertha Berlin peg back Stuttgart and earn a 2-2 draw after Omar Marmoush and Philipp Forster had netted early for the hosts.
Over in LaLiga, Rayo Vallecano were able to secure a 1-0 win over Espanyol to sit sixth in the table thanks to Leandro Cabrera’s 54th-minute own goal.
Valencia are eighth after they recovered from Iago Aspas’ early effort for Celta Vigo to clinch a 2-1 victory on the road following goals by Hugo Duro and Maxi Gomez.
Bottom side Levante remain without a win after a stalemate at home to Osasuna while Elche put more distance between themselves and the bottom three with a 3-1 triumph over Cadiz.
Rennes are Paris St Germain’s nearest challengers after a 5-0 success at St Etienne on Sunday saw them rise to second in Ligue 1.
Martin Terrier struck a hat-trick for the visitors, who next travel to face Tottenham on Thursday in the Europa Conference League.
Yvann Macon put through his own net and Lesley Ugochukwu wrapped up the scoring for Rennes but the thrashing signalled the end for Claude Puel at St Etienne after the former Southampton and Leicester boss was placed on gardening leave following the result.
Nice are still fourth but suffered a painful 3-0 reverse at home to sixth-placed Strasbourg. Ludovic Ajorque broke the deadlock in the 21st minute and efforts during the final 10 minutes by Habib Diallo and Adrien Thomasson wrapped up the victory.
Angers and Montpellier are also in the top half after narrow victories over Reims and Clermont respectively.
Lyon and Bordeaux played out a 2-2 draw and Nantes were able to get the better of Lorient 1-0.