Suppose a person wants to carry on gym work outs for an hour and the exercises he do in gym are only for upper body(hands and chest). Can running for 15 minutes will serves as a warm up for gym routine?
What running will do is increase your heart rate and blood flow which is good but it won't prepare the specific muscles your about to work on.
What I would advice is to do some dynamic stretching before working out any muscle group. You can also start the first exercise with 2 or 3 warmup sets. Let's say you're going to benchpress, first do 2 sets with about 50%-60% of the weight you'll be working with.
You can obviously combine all this with running for 15 minutes if you want to incorporate this in your warmup.