On the day it emerged Caixinha had banished veteran striker Kenny Miller from his first team squad amid the fall-out from their Old Firm defeat last weekend, the Ibrox men recovered from the loss of a first minute goal by Danny Redmond to claim an ultimately convincing win.
On-loan Welsh international Declan John’s double and a Daniel Candeias strike gave Rangers a 3-1 half-time lead and they then survived Ryan Jack’s sending-off and added to their advantage with a Graham Dorrans penalty.
Accies, whose long wait for a victory in this fixture continues, missed a spot-kick of their own in a dramatic and absorbing contest.
On an evening which many felt had the potential to fatally undermine Caixinha’s tenure, his team could not have made a more calamitous start.
He betrayed no emotion when Redmond put Accies ahead after just 57 seconds but must have been gripped by a sudden sense of dread.
Fabio Cardoso was the chief culprit for Rangers in the concession of the goal. The central defender was weak and indecisive as Rakish Bingham seized possession and surged into the penalty area.
The striker’s shot was parried by Wes Foderingham but only as far as Redmond who reacted quickest to force home a close range shot.
The visitors were stunned and initially appeared at a loss as to how to respond.
In the immediate aftermath of last Saturday’s 2-0 home defeat by Celtic, Caixinha had stated that this was when he would see what his team was made of.
It was through the purpose and energy of John that they gradually gave him the reaction he needed.
Accies had warning of how easily John was finding space down the left when he whipped over a fine cross which saw Xavier Tomas’ make a fine intervention to cut out the danger ahead of Josh Windass.
Carlos Pena, again struggling to get to the pace of play, was booked after just 14 minutes when his frustration manifested itself in a late challenge on Dougie Imrie.
The signs remained ominous for Caixinha but the picture changed dramatically with John’s first goal for Rangers rapidly followed by his second.
The 21st minute equaliser saw him drive forward and cut inside the flat-footed Ioannis Skondras before driving a precise right foot shot low beyond Darren Jamieson’s right hand into the corner of the net.
Four minutes later, Rangers were in front as the home defence again failed to track John’s adventurous movement.
This time, he played a clever exchange of passes with Windass before his left foot shot from the corner of the six-yard box beat Jamieson despite the goalkeeper getting a touch on the ball.
Accies were reeling and should have been further behind just a minute later. Ross McCrorie stepped out of defence and played a fine through ball to set Alfredo Morelos free on goal.
It was a simple chance for the Colombian striker but his shot was too close to Jamieson who managed to divert it wide.
Rangers’ third goal was merely delayed, however, as they increased their lead in the 27th minute.
Dorrans, captain for the night, lofted a clever pass over the top of the Hamilton backline to pick out the run of Candeias from the right.
The winger produced an emphatic finish, steering his right foot shot beyond Jamieson from a tight angle.
Hamilton, whose prospects of a first home win over Rangers for almost 80 years had all but evaporated with the quickfire loss of those three goals, struggled to rediscover the vigour with which they had started the match.
Redmond came close to giving them fresh encouragement before the interval when he worked himself into space for a shot from 20 yards which Foderingham did well to turn it behind with a smart save.
Accies’ hopes of a recovery appeared to be handed a boost with Jack’s dismissal just nine minutes into the second half.
Booked seven minutes earlier for a foul on Ali Crawford, the Rangers midfielder collected his second yellow card for a trip on Greg Docherty.
Caixinha responded with an immediate substitution, replacing Pena with Jason Holt as he reorganised his short-handed side. The change paid a quick dividend, Holt earning the penalty kick which allowed Rangers to give themselves the cushion of a 4-1 lead in the 59th minute. Holt was pulled back by Alex Gogic, leaving referee Andrew Dallas with a simple decision, and Dorrans coolly converted from 12 yards.
Bingham was far less convincing at the other end when he had the chance to pull one back for Accies after McCrorie fouled substitute Botti Biabi inside the box. Foderingham dived to his left to keep out Bingham’s penalty.
The Rangers ‘keeper made a brilliant save to keep out Steven Boyd’s close range header as the visitors saw the job through, providing Caixinha with a degree of vindication and some much needed breathing space.
Hamilton: Jamieson, Tomas, Gogic, McMann; Skondras, Docherty, Longridge (Biabi 63), Redmond (Donati 62), Imrie; Crawford (Boyd 76), Bingham. Subs not used: Fulton, Lyon, Hughes, Cunningham.
Rangers: Foderingham, Tavernier, Cardoso, McCrorie, John; Candeias (Dalcio 90), Jack, Dorrans, Windass; Pena (Holt 55); Morelos (Herrera 73). Subs not used: Alnwick, Hodson, Nemane, A. Wilson.