* Kirkuk flows to Ceyhan still at around 200,000 bpd
* U.S. rig count falls for third week - Baker Hughes
* Russia's Rosneft takes control of Kurdish oil pipeline
* Global refiners import U.S. oil to make up for OPEC cuts (Adds settlement prices, U.S. rig count, analyst quote)
NEW YORK, Oct 20 (Reuters) - Oil prices were up slightly on Friday in see-saw trade, ending the week up on support from a sharp decline in Iraqi crude exports due to tensions in the Kurdistan region after contending with weak demand data.
U.S. crude futures settled up 18 cents, or 0.4 percent, at $51.47 a barrel. Brent crude rose 52 cents, or 0.9 percent, at $57.75 a barrel. Both contracts were up slightly for the week.
The U.S. oil rig count fell for a third week in a row, extending a two-month drilling decline, General Electric Co's GE.N Baker Hughes energy services firm said.
"We've continued to see signs that the market needs a steady drumbeat of positive information," said Gene McGillian, director of market research for Tradition Energy. "This week's DOE report where gasoline demand dropped to its lowest since March gave a little pause to that."
Oil exports from Iraq's Kurdistan via the Turkish port of Ceyhan were flowing at average rates of 216,000 barrels per day (bpd), down from the usual flows of around 600,000 bpd, a shipping source said.
Iraqi troops regained control of two major oilfields northwest of Kirkuk from Kurdish Peshmerga forces this week, and the oil ministry in Baghdad expects to bring the fields back on stream on Sunday.
Russia's biggest oil company, Rosneft, has agreed to take control of Iraqi Kurdistan's main oil pipeline in a $1.8 billion investment.
Olivier Jakob, chief strategist at consultancy Petromatrix, said the deal with Rosneft "makes it a bit harder for Baghdad to do anything against those flows".
Analysts said the market was on a path toward rebalancing.
"The oil market has moved into modest undersupply and we expect this will persist at least through the end of the year," U.S. investment bank Jefferies said.
U.S. commercial stocks of crude oil have dropped 15 percent from their March records, to 456.5 million barrels, below levels seen last year. <C-STK-T-EIA>
Part of this drawdown has been due to rising exports as a result of the steep discount of U.S. crude to Brent, which makes it attractive for American producers to export their oil. <CL-LCO1=R>
Crude oil for immediate use now carries a premium over forward futures, making it profitable to sell oil produced now rather than storing it for sale later.
Shipping data in Thomson Reuters Eikon shows that overseas U.S. crude oil shipments have soared from virtually zero before the government loosened export restrictions in late 2015 to around 2.6 million bpd in October.
"While outbound shipments recently approached 2 million bpd, our math suggests that physical bottlenecks are unlikely to kick in until waterborne exports approach 3.2 million bpd," RBC Capital Markets said.
(Reporting by Ahmad Ghaddar; Additional reporting by Henning Gloystein and Vidya Ranganathan in Singapore and Christopher Johnson in London; Editing by Marguerita Choy and David Gregorio)