Boyd then put Gers ahead before substitute Nacho Novo made sure of the victory with a late clincher.
Having lost 1-0 to St Mirren in their last outing three weeks ago, it was the result Walter Smith was looking for from his men.
He’ll now be looking for more of the same when the teams do it all again at Ibrox in the League Cup quarter-finals this Tuesday.
Smith made two changes to the side which slumped to defeat at Love Street almost three weeks ago.
Charlie Adam was omitted from the squad completely while Jean-Claude Darcheville, still looking for his first goal of the season, dropped to the bench.
DaMarcus Beasley came in on the left side of midfield and Boyd started in his first appearance since making himself unavailable to play for Scotland.
Unsurprisingly, after a truly awful week of weasther in the west of Scotland, the game began in swirling gusts and driving rain.
Smith had warned his players they would come up against a Hamilton side which would treat this game like a cup tie.
But it was hard for both teams to settle in the opening stages, with the sodden surface limiting what they could do.
Rangers might have benefitted when a backpass was carried behind by the wind for a corner but Kirk Broadfoot fouled in the box when the ball came in.
Kevin Thomson almost put the visitors in front with an inswinging free kick from the right after Broadfoot was hauled down by Easton as he tried to cut in.
But after flashing past Accies keeper Tomas Cerny, the ball shot just past the far post as well and it remained 0-0.
Gers were beginning to control proceedings, with Martin Canning conceding another corner and Broadfoot nodding wide.
Canning was at fault once more when he pulled Boyd down 25 yards out and with sight of goal, the former Kilmarnock hitman fancied his chances.
His effort at the free kick left a lot to be desired, however, and his pulled drive was always trundling away.
Back came Hamilton, who served warning they weren’t just making up numbers when a centred ball rebounded off the crossbar from a ruck of bodies.
Gers didn’t take heed and, after 25 minutes, the Lanarkshire outfit went in front in similar circumstances.
Simon Mensing swung a free kick in from the left to the back post and when Easton connected, it evaded Allan McGregor and dropped over the line.
Rangers’ response wasn’t as emphatic as manager Smith would have liked and although they had the wind at their backs, that actually hindered them.
The sight of a through ball from the away defence was all too common as conditions played an increasing part in how the game panned out.
Davis tried to draw Gers level and came close when his curling effort from 25 yards whistled inches wide.
Trent McClenahan sounds more like an American political advisor than a Premier League footballer and it was his mistake that finally gave the visitors a way back into the match.
His foul on Kenny Miller in the penalty area meant referee Dougie McDonald pointed to the spot and Boyd made no mistake from 12 yards four minutes before the break.
Rangers set about trying to take a lead after the interval and within four minutes they had it.
Boyd was the scorer again, sending a simple header into the net after Broadfoot released Davis on the right and the Northern Ireland midfielder crossed.
The striker nearly grabbed his hat-trick when Miller scrambled into space on the right but his he couldn’t connect with the low cross.
Boyd then tried to turn provider for his attacking colleague after good work by Davis but the move came to nothing.
Madjid Bougherra tried to get in on the act with a header, yet it never looked like sneaking in from Thomson’s corner and flew well over instead.
Hamilton made a double substitution as they tried to salvage something and Billy Reid threw on David Graham and Mark Corcoran.
The Light Blues also made a change, with Novo – sporting bright red boots – taking the disappointing Beasley’s place.
Boyd again looked like clinching a treble as the game lurched into the closing quarter but his placed effort was deflected off target and he headed wide seconds later.
It seemed only a matter of time before Rangers would make the points safe and Davis was next to stab towards goal.
Once again, there was an unsuccessful outcome but with 11 minutes remaining Novo did make it 3-1.
A shocking mistake by Canning let the little forward in and when he slotted under Cerny, he showed why he’d be a good addition to the Scotland squad should he be called up.
Gers tried to add extra gloss to their triumph but although they couldn’t find the net again, the result was never in much doubt once they edged in front.