Joe Aribo scored Rangers' second goal.
Rangers held off Livingston's fightback to maintain their lead in the Scottish Premiership.
Scott Arfield and Joe Aribo scored goals in the first 16 minutes to put the visitors in control.
Fashion Sakala killed off their challenge after Bruce Anderson responded for Livingston.
Rangers are six points ahead of Hearts and seven ahead of Celtic.
The chance to get back to four points is present when they face Aberdeen on Sunday.
Reaction as it happened.
Steven Davis was replaced by Arfield in the middle of the game as Van Bronckhorst made just one change for his first match in charge.
The Canada international ran in behind the Livingston defence to collect James Tavernier's pass and score the first goal of the game.
Ryan Kent kept his width on the left flank and he was able to receive a diagonal switch before feeding Morelos, who set up Aribo to sweep a brilliant finish off the under side of the goal.
Livingston were able to take advantage of Rangers' slackness at the back. Jack Fitzwater arrowed a long ball in behindConnor Goldson for Alan Forrest, and Anderson pounced to head home when his team-mate's shot was saved.
Forrest's back-post shot was pushed over the bar by Allan McGregor before the break as Borna Barisic stood off and allowed Bailey's cross to arrive.
After a 10-minute delay due to Rangers fans throwing snowballs on the pitch, both teams started to play better after the break and Stryjek had to make a number of saves.
Livingston was in the fight until the final 12 minutes when substitute Sakala capitalised to head in Rangers' third.
Ryan Kent was the man of the match.
Kent was influential as he stayed wide, and looked sharp after his return from injury. Tavernier had two assists to show his consistency.
What did we learn?
There were a few hints of change from Thursday, but Van Bronckhorst has picked up from Steven. The use of Kent and Hagi as more traditional wingers, rather than the usual number 10s, was evident again in the first period, and the former's width was crucial to the second goal as switches of play proved effective.
Hagi picked up positions between Livingston's centre-backs and left-wing backs and looks like he will get plenty of time to build consistent form under the new manager. It's possible that the Scotland full-back may be in for more game time as well.
Rangers have conceded more goals in their first 14 league games than they did in the whole of last season. Solidity remains a problem even if last term was freakish.
Livingston's goal came from a diagonal ball into the space between Goldson and the advanced Tavernier, and the former was erratic in possession on a few occasions in the first half. The Croatian was more influential in the attack after the break because Barisic stayed back more often.
David Martindale was marking a year in charge of Livingston and he was pleased with his side's efforts, despite falling two goals behind.
Anderson put in a great shift up front and Stephane Omeonga battled well in the middle. The manager would have liked to see more from his team in attack as they faded after a fast start to the second half.
What's next?
Both sides have away trips in December. Rangers and Livingston will play at the same time on Wednesday.
Line-ups.
Match statistics.
Live text.
Formation 3-4-2.
21McMillan18Holt33Omeonga
32Stryjek
The book was written at 45 minutes.
5 Fitzwater.
4 parks
Substituted forriceat 59' minutes.
18Holt
Booked at 69 minutes.
rest booked at 31 minutes
Substituted for Kabia at the end.
11Montao
Substituted for Shinnie at 59 minutes.
Substitutes.
3 Longridge
10Sibbald.
15Panayiotou
Shinnie.
29Penrice
36 Maley.
37 Kabia.
Formation 4-2-3-1.
2Tavernier6Goldson3Bassey31Barisic
McGregor is a person.
2Tavernier
6Goldson.
3Bassey.
31Barisic
18Kamara.
17Aribo
Substituted forPatterson at 73 minutes.
Booked at 9 minutes.
Substituted forSakalaat 73 minutes.
Substituted forDefoeat for 90 minutes.
Substitutes.
4Lundstram
9Defoe
10Davis
16Patterson
23Wright.
30Sakala
McLaughlin is 33
The match ended with Livingston defeating Rangers.
Livingston won the second half by a score of 1.
The Rangers' Glen Kamara was the victim of a foul.
Holt won the free kick in the defensive half.
The attempt was saved. Jermain Defoe's shot from outside the box is saved in the bottom left corner. Scott Arfield assisted.
The Rangers' Nathan Patterson was the victim of a foul.
Andrew Shinnie won a free kick.
The Rangers had a substitution. Jermain Defoe is the new player.
The attempt was saved. Scott Arfield's shot was saved in the bottom left corner. They were assisted by Alfredo Morelos.
The attempt was blocked. The shot from outside the box is blocked by the other team.
James Tavernier of the Rangers.
The left wing is where the free kick was won by Montao.
Joe Aribo won a free kick.
Stephane Omeonga had a foul.
It was Livingston who was trying. Jack Fitzwater tries a through ball, but Jaze Kabia is not.
The attempt was saved. The left footed shot from outside the box is saved in the bottom right corner. Glen Kamara assisted.
The substitution was Livingston. Jaze Kabia is the new person.
Goal! Livingston defeated the Rangers. Sakala headed from very close range to the high centre of the goal. James Tavernier assisted with a cross.
Corner, Rangers. Tom Parkes was Conceded.
The attempt was blocked. Sakala's shot from very close range is blocked. They were assisted by Alfredo Morelos.
