* U.S., European stocks rise, Asian stocks fall
* Euro falls further on Thursday's ECB news
* European bank stocks snap rally
* Oil rises, U.S. treasury yields higher (Updates to late afternoon trading)
By Sinead Carew
NEW YORK, Dec 9 (Reuters) - U.S stocks followed Europe higher on Friday as investors piled in on the post-Trump-election rally while the euro weakened against the dollar as investors continued to digest the European Central Bank's Thursday decision to extend economic stimulus.
U.S. Treasury yields climbed, with benchmark yields on track for a fifth straight week of increases on stronger-than-forecast data on China inflation and U.S. consumer sentiment ahead of $56 billion in government debt supply next week.
European shares hit their highest level and had their best week since January, following the ECB's decision to cut monthly bond buys to 60 billion euros ($63.7 billion) from 80 billion and extend purchases to December, three months longer than analysts had forecast.
The S&P 500 stock index's strongest sectors were consumer staples and healthcare, which have been two of the weakest in the rally that followed the Nov. 8 presidential election.
"Today we're seeing money going into some of the lesser loved sectors since the election, which is telling me the rally is broadening which is a very positive sign," Randy Frederick, vice president of trading and derivatives for Charles Schwab in Austin, Texas.
While some investment managers may be taking their profits others who did not trust the post-election rally would last are likely now hoping to show gains on their books for the year-end by picking up sectors that look less expensive.
"It means there's still new money coming in. People are worried about getting left behind at this point," he said.
The Dow Jones industrial average was up 89.87 points, or 0.46 percent, to 19,704.68, the S&P 500 had gained 8.64 points, or 0.38 percent, to 2,254.83 and the Nasdaq Composite had added 16.64 points, or 0.31 percent, to 5,433.99.
The euro fell for the second day after the ECB news gave the euro zone common currency its biggest daily loss against the dollar on Thursday since Britain's vote to leave the European Union in June.
It traded around $1.0541, down 0.7 percent on the day and down 1.2 percent for the week. The dollar was up 0.5 percent on the day up 0.8 percent for the week against a basket of major currencies.
"The extension of the program was longer than most had expected, and a lot of the language ... was pretty dovish," said Erik Nelson, a currency analyst at Wells Fargo in New York. "Inflation forecasts were pretty subdued all the way out to 2019, and growth forecasts were pretty low, and the risks are tilted to the downside."
Europe's STOXX 600 finished up almost 1 percent even though European bank stocks pulled back 0.7 percent. Reuters reported that the ECB rejected a request from Italy's Monte dei Paschi di Siena bank for more time to raise cash. The continent's banking sector was still up 9.5 percent for the week, its biggest jump since December 2011.
Oil rose on growing optimism that non-OPEC producers might follow the cartel's lead by agreeing to cut output.
U.S. crude settled up 1.3 percent at $51.50 a barrel. Brent crude settled up 0.8 percent at $54.33.
Spot gold was down 1 percent and was set for a weekly decline of 1.5 percent, pressured by the stronger U.S. dollar and expectations of a Fed rate hike. (Additional reporting by Abhinav Ramnarayan in London; Editing by Nick Zieminski and James Dalgleish)