* Spot Eonia falls to record low
* ECB rate cut, liquidity injection still to take effect
* Forward breakevens rise with inflation outlook
* Greek yields drop, Cyprus plans comeback (Updates prices, adds new quotes)
By John Geddie and Marius Zaharia
LONDON, June 10 (Reuters) - Euro zone overnight bank-to-bank interest rates traded at record lows on Tuesday and short-term bonds outperformed longer-dated paper in a sign that the European Central Bank's efforts to keep money markets anchored may bear fruit.
A measure of the market's inflation expectations, derived from the difference between the yields of index-linked debt and conventional bonds - the euro five-year, five-year breakeven forward - was near three-month highs, showing increasing confidence in the ECB's resolve to accelerate price growth.
While it was still early days, strategists said the market's response to last week's cut in the ECB's main interest rates and its promise of fresh liquidity for banks suggested the measures could work. The moves could go further in the coming days as the ECB measures come into effect.
"Everything that has happened in the market is in line with what the ECB would have hoped for," said David Keeble, global head of fixed income strategy at Credit Agricole. "The ECB is getting in front of the curve again."
Spot Eonia fixed at 0.053 percent after markets closed on Monday, a new historic low.
The five-year, five-year breakeven forward rate , which is one of the ECB's favourite tools for gauging the market's inflation expectations, traded around 2.13 percent, some 3 basis points up from before the ECB meeting.
It moved further away from lows of just above 2 percent hit in May after below-forecast inflation data.
The amount of cash euro zone banks have beyond what they need for their day-to-day operations is a key factor holding short-term rates low. Excess liquidity stands at 120 billion euros, well above the three-year low of 70 billion euros hit at the end of last month.
A smaller take-up from banks at the ECB's weekly liquidity offerings on Tuesday will squeeze 25 billion euros out of the banking system for the coming week, but the impact is expected to be limited and in any case temporary.
From next week, the ECB will inject around 170 billion euros into the banking system by halting tenders that withdrew funds spent on past government bond purchases.
It has also introduced 400 billion euros of ultra-cheap four-year loans for banks - conditional on their lending to the smaller companies that are Europe's economic backbone - which will be available from September.
ECB Governing Council member Erkki Liikanen reiterated on Tuesday that the ECB still has tools it can employ if needed.
Forward Eonia rates are already pricing in expectations the ECB will keep the market pinned down, while six-month contracts dated for 12 months from now are low enough to suggest further policy loosening, say strategists.
Investors' preference for short-dated German bonds over longer-dated ones is further evidence of confidence in what the ECB can achieve with its latest measures.
Two-year German yields fell slightly to 0.06 percent. Ten-year yields, the benchmark for euro zone borrowing costs, rose 2 bps to 1.40 percent.
"The ECB has given very strong forward guidance for the first two years, and all its measures work to pin front-end rates," said Michael Michaelides, rates strategist at RBS.
The ECB's pledge to keep rates at historic lows for some time has, however, fuelled demand for the higher-yielding bonds of some of the bloc's weakest members.
Yields on 10-year Greek bonds dropped to as low as 5.48 percent, a level not seen since January 2010, while Portuguese equivalents were within a whisker of euro era lows after dropping 14 bps to 3.24 percent.
These new lows raise the prospect that Greece, which returned to markets in April for the first time since 2010, could soon issue more debt to help to stave off the need for a third bailout, Commerzbank said in a note.
The Republic of Cyprus - bailed out just last year - is set to meet investors ahead of a possible euro bond issue, IFR reported. (Editing by Catherine Evans and Nigel Stephenson)