Steyn weighed in with two tries and was at the centre of Glasgow's best moments after a slow start. Smith told the Sunday Times.
"He [Steyn] played a captain's innings today and I'm really proud of that because we were underperforming in the first 10-15 minutes and I asked him specifically - I got a message onto him - I told him you have to put in an important shift and be a point of reference for us," Smith told the Sunday Times.
"He played really well and through actions he showed his leadership. Sometimes you can talk things right, sometimes you can motivate other people but to play things right is a special ability and I'm really proud of him."
Warriors took their time to acclimatise and Smith acknowledged that the warm conditions suited Zebre better than them.
"The temperature was -1C when we woke up on Friday morning in Glasgow and it was more than 20C here in Parma so I can understand why we looked a little bit lethargic," said the coach. "That's not an excuse, obviously, but we needed to tire these guys out. They were very well motivated and very well prepared. Because we stuck to the plan and the process, we got the result."