Rangers suffered their heaviest defeat at the national stadium to their great rivals since the 1969 Scottish Cup final - also a 4-0 thrashing.
Celtic are now one win away from two ideal seasons where they completed the treble of the league, League Cup and Scottish Cup.
Ross McCrorie was then sent off shortly after half-time and Moussa Dembele scored the resulting penalty before Olivier Ntcham wrapped up the win with another spot-kick.
However, that was to comeback to haunt them at Hampden as they were dispatched by four goals as Celtic booked a final showdown with Motherwell next month. We are in a really good position there, but have five tough games.
"Then in the second half, we gave them a penalty and a sending-off which makes their job far, far easier".
"I was reacting to the situation in the game".
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Nemanja Matic panicked at the delivery and headed back inside to Rodriguez , who could not miss from three yards out. United manager Jose Mourinho had long given up hope of winning the title, instead focusing on finishing second.
"The first half was a total embarrassment for Rangers, they were so far off the pace". There are things we can all get better at but it's too raw right now to go into it.
Victory over Hibs at Easter Road on Sunday would secure a seventh successive title but, reluctant to become involved talk of back-to-back Trebles, Rodgers claimed: 'I haven't really thought about that.
As a coach and manager you are always looking for that difference before the game, the one per cent, but I said to the players that the edge is them, added Rodgers. "We have to put a performance in that is worthy of their support".
"There was emotion, anger, disappointment in our dressing room - all the things you would expect after a game like that". We were just passive.
There could be difficulties in agreeing a new deal though, as Australian football website The World Game reported in February that the player had turned down a new deal with the Hoops, although this was denied by Rodgers, as reported by Fox Sports Australia. I don't think we got close enough to them. "It is about making sure they turn that into positives and energy to power us through to the end of the season, rather than allowing it to be disruptive".