But investors are warily watching Ukraine as diplomats try to avert a civil war.
Sydney gained 0.35 percent, or 19.2 points, to close at 5,481.4 and Taipei added 0.47 percent, or 41.96 points, to 8,912.39. Wellington fell 0.49 percent, or 25.36 points, to 5,174.90.
Shanghai ended flat, edging up 0.69 points to 2,028.04, and Manila was also almost unchanged, dipping 0.53 points to 6,765.93.
Tokyo, Hong Kong and Seoul were shut for public holidays.
New York dealers provided a healthy cue, with all three of Wall Street's main indexes advancing after the Institute for Supply Management said its purchasing managers index (PMI) for the US services sector rose to 55.2 in April from 53.1 in March.
It was the highest reading since last July and better than the 54.0 figure expected by analysts. A number above 50 indicates expansion in the sector, which accounts for the bulk of US economic activity.
Figures released Friday showed ISM's manufacturing-sector PMI rose for the 11th straight month in April, to a better-than-expected 54.9.
The Dow edged up 0.11 percent, the S&P 500 added 0.19 percent and the Nasdaq was 0.34 percent higher.
However, events in Eastern Europe are keeping traders on edge.
Fighting intensified between government forces and pro-Moscow secessionists on Monday, with Ukraine's military suffering heavy casualties and Russia warning that violence was endangering peace in Europe.
Ukraine's interior ministry said pro-Russian gunmen controlling the town of Slavyansk were using civilians as human shields and shooting from houses, some of which were ablaze.
Diplomats scrambled to ease the worst East-West crisis since the end of the Cold War, with UN Secretary General Ban Ki-moon offering himself as a personal go-between.
Speaking exclusively to AFP in Abu Dhabi, Ban offered "to provide my own role if necessary" before the crisis "spins out of control and creates huge consequences beyond anybody's control".
European leaders, fearing all-out civil war on their eastern flank, have also launched a new peace bid, urging Ukraine and Russia to find a negotiated solution.
On foreign exchange markets the dollar bought 102.03 yen against 102.14 yen in New York on Monday, while the euro was at $1.3900 and 141.90 yen, compared with $1.3874 and 141.72 yen.
The Australian dollar rose to 92.99 US cents from 92.72 cents after a widely expected decision by the Australian central bank to keep interest rates on hold.
Oil prices edged up. The US benchmark, West Texas Intermediate for June delivery, rose 10 cents to $99.58 a barrel in afternoon trade while Brent North Sea crude for June added four cents to $107.76.
Gold fetched $1,311.77 at 0735 GMT compared with $1,308.83 on Monday.