The duo scooped the accolades for their role in helping the Ibrox club win all five of their league matches during a busy period for the club.
The month started with a 3-0 victory over Dundee and was followed by back-to-back 2-0 triumphs over Hearts and St Johnstone before rounding off with wins over Dundee United and St Mirren.
Speaking after he picked up his award, a delighted Van Bronckhorst said: “I think the start of my career at Rangers has been very positive, and I am very proud to win this award, but I think what is behind the award is much more important.
“It means we have done a great job and my staff who have come with me and the staff who are already here have helped me a lot and have worked hard to achieve the wins we have had.
“Also the players - I thank them as well for all the work they have done in the last weeks and month, and that is why I have got this award.
“It means you are doing a good job and gives me more motivation to kick-on.”
Colombian striker Morelos enjoyed a successful month after a sluggish start to the season, netting four goals.
His first arrived against Dundee before opening the scoring eight days later against the Jambos at Tynecastle.
Morelos then bagged the breakthrough goal against St Johnstone and rounded off his spree with the clincher over St Mirren on Boxing Day.
Van Bronckhorst heaped praise on his talisman, stating: “He has done well since I came in and he has scored some important goals that have been important for the team.
“He is working hard and his attitude has been really good, and he has been a positive factor for the team.”
Morelos has been linked with a January move to Newcastle United and is also interesting Leicester City and Everton, but it will reportedly take an offer in the region of £20million to prise him away from Ibrox.
The Magpies, who are involved in a Premier League relegation battle, are actively searching to improve their attacking options with lead striker Callum Wilson currently side-lined due to injury.
Since joining Rangers in June 2017 from HJK Helsinki, Morelos has netted 68 goals in 138 appearances.