Westwood replaced the Manchester City keeper on Friday night at half-time of the Ireland's 1-1 draw with Nigeria at Craven Cottage.
"He's a very good goalkeeper and has impressed me in training," Given said.
"He'll be the future Irish goalkeeper, for sure, pushing harder in the next couple of years and then after that I'm sure he'll get the mantle."
With former Sky Blue Dean Kiely walking out of the squad, Westwood is expected to be on the bench for the World Cup qualifier in Bulgaria on Saturday.
Ireland will open their new Aviva Stadium in Dublin on August 11 2010 with a friendly against Argentina. It will also stage the 2011 Europa League Cup Final.