Emerson Hyndman converted James Tavernier's cross to put the hosts ahead and Clint Hill knocked in Jon Toral's delivery for their second.
Martyn Waghorn tucked away from the penalty spot after being fouled by Hamilton's Massimo Donati.
And Ibrox captain Lee Wallace's strike completed the scoring.
The Ibrox side are eight points behind second-placed Aberdeen but are seven clear of St Johnstone, with third place guaranteeing a Europa League qualifying spot.
Accies fall below Inverness Caledonian Thistle on goal difference and are three off Motherwell.
Caixinha, 46, had signed a three-year deal to take over from Mark Warburton earlier this month but took in Sunday's 1-1 draw with Celtic from the stand while Graeme Murty continued in interim charge.
There was little change in personnel on the field as the Portuguese took charge in the dugout, with Toral replacing Lee Hodson while Hamilton player-manager Martin Canning made three changes following last week's 4-0 loss to Hearts.
Rangers' intent early on was clear as Jason Holt and Waghorn had efforts before on-loan Bournemouth midfielder Hyndman finished off an inventive move involving Kenny Miller and Tavernier.
And, from Toral's whipped free-kick, Hill continued his recent scoring run to take his season's tally to six from close range.
Former Celtic player Donati felled Waghorn and the Englishman scored his fifth goal in five games against Accies this season.
And Waghorn turned provider for Wallace's goal, the Scotland left-back driving home from a tight angle.
Accies, who had brought Alex Gogic, Grant Gillespie and Darian MacKinnon into their starting line-up, rarely threatened and have now lost 14 goals in their last three games.
The Premiership breaks for international football next weekend and Rangers resume their campaign at home to Motherwell on 1 April, when Hamilton host St Johnstone.
Match ends, Rangers 4, Hamilton Academical 0.
Second Half ends, Rangers 4, Hamilton Academical 0.
Foul by Joe Garner (Rangers).
Georgios Sarris (Hamilton Academical) wins a free kick in the defensive half.
Corner,  Hamilton Academical. Conceded by Wes Foderingham.
Corner,  Hamilton Academical. Conceded by Emerson Hyndman.
Corner,  Rangers. Conceded by Alexander Gogic.
Corner,  Hamilton Academical. Conceded by Jason Holt.
Lee Wallace (Rangers) is shown the yellow card for a bad foul.
Foul by Lee Wallace (Rangers).
Alejandro D'Acol (Hamilton Academical) wins a free kick on the right wing.
Substitution, Hamilton Academical. Gramoz Kurtaj replaces Daniel Redmond.
Substitution, Rangers. Harry Forrester replaces Kenny Miller.
Foul by Jon Toral (Rangers).
Scott McMann (Hamilton Academical) wins a free kick on the left wing.
Giannis Skondras (Hamilton Academical) wins a free kick on the left wing.
Foul by Joseph Dodoo (Rangers).
Jason Holt (Rangers) wins a free kick on the right wing.
Foul by Darian MacKinnon (Hamilton Academical).
Attempt missed. Joe Garner (Rangers) right footed shot from the centre of the box is close, but misses to the right.
Corner,  Rangers. Conceded by Massimo Donati.
Joe Garner (Rangers) wins a free kick on the left wing.
Foul by Alexander Gogic (Hamilton Academical).
Substitution, Rangers. Joseph Dodoo replaces Martyn Waghorn.
Substitution, Hamilton Academical. Giannis Skondras replaces Grant Gillespie.
Goal!  Rangers 4, Hamilton Academical 0. Lee Wallace (Rangers) left footed shot from the left side of the six yard box to the centre of the goal. Assisted by Martyn Waghorn.
Attempt blocked. Dougie Imrie (Hamilton Academical) left footed shot from outside the box is blocked.
Substitution, Hamilton Academical. Alejandro D'Acol replaces Greg Docherty.
Foul by Jason Holt (Rangers).
Dougie Imrie (Hamilton Academical) wins a free kick in the attacking half.
Substitution, Rangers. Joe Garner replaces Barrie McKay.
Attempt saved. Kenny Miller (Rangers) left footed shot from the centre of the box is saved in the bottom right corner.
Jason Holt (Rangers) wins a free kick in the attacking half.
Foul by Dougie Imrie (Hamilton Academical).
Foul by Jon Toral (Rangers).
Daniel Redmond (Hamilton Academical) wins a free kick in the attacking half.
Corner,  Rangers. Conceded by Alexander Gogic.
James Tavernier (Rangers) is shown the yellow card for a bad foul.
Foul by James Tavernier (Rangers).
Daniel Redmond (Hamilton Academical) wins a free kick in the defensive half.
Rangers marked Pedro Caixinha's first match as manager with a comfortable Scottish Premiership win over Hamilton Academical, who slipped to bottom.