The Federal Reserve has - as expected - decided to keep US interest rates on hold at 1%.
Inflation pressure could emerge, Mr Greenspan said
But analysts now expect that the Fed will begin raising rates in the summer, in an attempt to match the strongly accelerating growth of the US economy.
In its accompanying statement the Fed itself hinted at a future upward trend, saying rates could now be boosted at a "measured" pace.
The news sparked falls in the dollar, taking the currency to a four-week low.
The Fed last raised US interest rates from their 1958 low back in 2000.
US stocks immediately shot up on the announcement, with the Dow Jones blue-chip index of leading shares rising by 47.3 points to 10,361.3, before falling back to end up 3 points to 10,317.
Against the euro in Asia, the US dollar dropped as far as $1.2140, having already fallen on Wall Street the previous evening.
"At this juncture, with inflation low and resource use slack, the Committee believes that policy accommodation can be removed at a pace that is likely to be measured," its Federal Open Market Committee said in its accompanying statement.
Previously it has called for "patience" before any future rise in interest rates.
Analysts welcomed the Fed's announcement, with the general consensus being that they were pleased the bank said future rises would be measured rather than rapid.
"They are saying that interest rates must go up at some stage but they will go up in
a very gradual way," said Rick Egelton, deputy chief economist at the Bank of Montreal.
"When they say the pace is 'likely to be measured' they are saying that they will move the rate up in gradual, steady increments."
The Federal Reserve has cut borrowing costs 13 times in the last three years in an effort to offset the impact of a global economic downturn, and the threat of terrorist attacks.
But recent data pointing to strong economic growth and a big rise in recruitment by US companies looks likely to prompt the Fed to change its stance.
Last month, figures highlighting a sharp increase in the number of new jobs created in March helped restore confidence in the US recovery.
Until then, persistently sluggish rates of new job creation had fuelled fears that worsening employment prospects could dent consumer spending, depriving the economy of one of its main growth drivers.
Any summer increase in interest rates could however curb stock market gains and weigh on spending by heavily-indebted US consumers in the run up to the November presidential elections.