The Bianconeri were held to a 1-1 draw by Venezia in their previous top-flight outing but never looked in danger of dropping more points after Morata's early opener.
Massimiliano Allegri's side largely toiled in the second period, yet Cuadrado ensured there was to be no nervy finish with a deflected strike in the 69th minute.
The result lifted Juventus up to sixth, level on 31 points with fifth-placed Roma, which shock Atalanta 4-1 earlier in the day, while Bologna remains in 10th.
Juve went ahead in the sixth minute when Morata lashed past Lukasz Skorupski from six yards after being superbly picked out by Federico Bernardeschi.
Mattias Svanberg went close with an audacious overhead kick midway through the first half as Bologna responded well to that early setback, though it could not find a leveller before the interval.
Juve offered little in the way of attacking threat immediately after the break, only making sure of the win when Cuadrado's powerful strike from just inside the penalty area was deflected past Skorupski by Aaron Hickey.
That goal freed up the visitors to push forward in the closing stages, with Bernardeschi's low strike tipped wide by Skorupski, yet they ultimately failed to add further gloss to the scoreline.
Juve will wrap up 2021 with a home game against Cagliari on Wednesday (AEDT), while Bologna travels to Sassuolo a day later before the mid-season break.