Boyd gave Rangers the lead against Falkirk at Ibrox
Rangers cut Celtic's lead at the top of the Scottish Premier League to two points after beating Falkirk at Ibrox.
Steve Lovell put the Bairns in front after 12 minutes, but Rangers levelled with a disputed Kris Boyd penalty shortly before half-time.
Falkirk's Patrick Cregg and Rangers' Sasa Papac were both sent off before Boyd's tap-in gave Rangers the lead.
And Steve Davis sealed the home side's victory in injury time with a well-taken strike.
The Bairns had broken the deadlock when Lovell got in behind Madjid Bougherra to latch on to Neil McCann's pass and finish off the outstretched foot of goalkeeper Allan McGregor.
Errors by Falkirk goalkeeper Robert Olejnik and defender Lee Bullen almost gave the hosts a quick equaliser, but Rangers were unable to take advantage on each occasion.
Olejnik tipped over from Kenny Miller's shot as the Falkirk defence, marshalled by debutant and former Rangers and Celtic player Steven Pressley, continued to absorb pressure.
Rangers claimed for a penalty when Boyd went down in the box under Olejnik's challenge, but the goalkeeper was adjudged to have made contact with the ball as he conceded a corner.
Moments later, Falkirk failed to clear under pressure and Kirk Broadfoot played Miller in on goal, but the striker's shot flashed wide.
Referee Willie Collum denied further spot-kick appeals by Barry Ferguson and Miller before Papac tested Olejnik from distance.
But Rangers were awarded a penalty when Cregg was penalised for a challenge on Boyd and Bullen was booked for dissent as Falkirk protested.
Boyd blasted past Olejnik from 12 yards.
McGregor had to come out of his penalty area early in the second period to clear as Lovell chased a long ball and, with the goal empty, Scott Arfield's speculative shot from near the half-way line went over.
Former Rangers defender Pressley (left) was making his debut for Falkirk
Cregg was sent off after 51 minutes for what referee Collum regarded as a dangerous tackle on Pedro Mendes near the edge of the Falkirk box.
John Fleck, who was making his first start for Rangers, sent over an excellent cross from which Boyd headed into the arms of Olejnik.
Another Fleck run set up a chance for Miller, but the Scotland striker's tame shot was held by the Austrian keeper.
Mendes crossed for Broadfoot to head towards goal and Olejnik was at full stretch as he tipped round the post.
With 13 minutes remaining, Papac was given a straight red card for a cynical tackle on Lovell's replacement Mark Stewart as Falkirk tried to mount a counter-attack.
Rangers went in front three minutes later when Olejnik blocked substitute Nacho Novo's low cross and Boyd tapped home the loose ball.
Higdon, who replaced Pressley, was almost put in on goal as Falkirk came forward, but a combination of David Weir and Broadfoot ushered the ball back to McGregor.
Stewart then headed wide at the back post from a deep cross.
Davis got Rangers' third when he came in from the left and escaped the challenges of Kevin McBride and Dean Holden before firing low past Olejnik.
Celtic can restore their five-point advantage when they play Aberdeen on Sunday.
Rangers manager Walter Smith: "We're delighted with the three points.
"We lost a rather slack goal. That gave them an extra impetus, being a goal up. We managed to pin them back and get a penalty on half-time.
"In the second half, we started well enough but couldn't get the breakthrough and then we managed to get the second goal.
"We had a couple of hairy moments before we got the third goal."
Falkirk manager John Hughes: "I feel for the boys, they gave me everything. They were outstanding.
"I just felt that when we scored, we stopped passing the ball and sat too deep.
"The goal was a bonus, it gave us plenty to hold on to but the penalty just before half-time was soul-destroying.
"Let's not take nothing away from Rangers. We were camped in and they were patient."
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