The 28-year-old Mitchelton-Scott rider launched a solo chase on race leader Deignan on the final climb of the UCI Women's WorldTour race, hauling back 23 seconds on the Trek-Segafredo rider on the descent into Liege, to finish nine seconds behind.
Brown dropped the rest of the chasing group on the final climb, the Cote de la Roche-aux-Faucons, to try to reel in Deignan. She had the British rider in sight several times on the twisty run-in but could not close the gap completely.
"It was a real time trial and I know that I’m a stronger time triallist than Lizzie Deignan," Brown said. "But it was hard when the majority of road left was downhill, you have to put out a lot of power to gain anything there.
“Part of your head is like ‘ah I can’t do it’, then the other part is saying; ‘you’re so close, you just need to push harder’.
"Bittersweet, but really excited to have my first WorldTour podium."
Deignan's team mate, Dutch rider Ellen van Dijk finished third.
Deignan, who added Liege-Bastogne-Liege to wins in the GP Plouay and La Course, was relieved to hold off the fast-finishing Aussie for her third victory since the season resumed.
"It was really horrible," Deignan said. "In the end, I decided to try to chase the motorbikes in front of me rather than think about her behind me, because mentally it is just cracking you when you think like that.
"I knew that (Brown) would descend faster than me, so I was just praying the line would come quicker."
Brown's second-place finish lifts her to 13th overall in the Women's World Tour rankings for 2020.