The Millers saw their three-game winning run on the road end at the Vitality Stadium as Philip Billing’s first-half goal proved decisive.
But Warne's side more than matched the Cherries, who look close to appointing Thierry Henry as their new manager, and could easily have left with a point.
Richard Wood, Freddie Ladapo and Ben Wiles all had good opportunities but despite a late barrage, the Millers made the lengthy trip home empty handed.
“We were OK. We were pretty pleased how we played,” Warne said. “We were good out of possession but not so good out of possession.
“The lads in the dressing room are a bit disappointed we have come away with nothing, which is a good sign.
“We threw the kitchen sink and taps at them as best we could and the best compliment that I can pay the lads is that Bournemouth were quite happy to hear the final whistle.
“I am really proud of the team because we gave a good account of ourselves.
“I thought we created a lot and made them defend a lot so our effort and endeavour should be applauded.
“I came here with confidence that we would cause them problems if we played at our best and we go away full of confidence.
“We have shown how well we can play and that we can be competitive against anyone.”
The Millers returned home in the early hours of Thursday morning and barely 36 hours later they will be hitting the road again as they visit league leaders Norwich on Saturday.
It is a game that has emotional attachment for Warne, who used to visit Carrow Road as a boy and still speaks of his love for the Canaries.
To help us maintain the high-quality reporting that you are used to from the football team at The Star, please consider taking out a subscription to our sports-only package.
You'll find all the details at https://www.thestar.co.uk/subscriptions/sports.
Your support is much appreciated.