Oldham, who was part of the British male team that won a first Olympic medal in the event since 1912 at London 2012, executed a highly difficult routine superbly to score 15.133.
That gave him silver and Britain's first ever medal in a major championships on the high bar while Emin Garibov took gold with 15.433 and Aliaksandr Tsarevich bronze with 14.833.
Double Olympic bronze medallist Whitlock became the first British male gymnast to win the European floor title earlier in the Championships to add to his all-around silver.
Whitlock also claimed bronze on the pommel horse with British teammate Daniel Keatings reclaiming his gold from 2010 in the same event in Russia.
And Oldham admitted it was about time he claimed an individual major championship medal after getting in on the act and completing Britain's success in Moscow.
"It's amazing it really is, it's been coming a long time, I was fourth in this event a few years ago and have been waiting to come back and prove myself," said Oldham.
"The first high bar medal for Britain, it's amazing. I've been working high bar for a long time and I really am so happy to have this silver medal around my neck.
"I knew on landing that I'd done a good routine and I would be close but it's not until you see your name on the scoreboard that you believe it, it's brilliant."
Fellow Brit Ashley Watson joined Oldham in the high bar final, finishing eighth, while Ruby Harrold place sixth in the women's beam final.