Also moving on into the second round was Scotlands Colin Montgomerie, who defeated the lone American in the event, Fred Funk, 3-up.
Vijay Singh advanced past the first round with a 1-up win over young Englishman Justin Rose.
In the days first match out but last on the course, Nick Faldo and New Zealands Michael Campbell were all square through 42 holes before play was suspended due to darkness.
The format has each pairing playing 36 holes all the way through the finals.
Harrington trailed most of the morning 18 but was able to get the match to all square by lunch. He then took control on the 26th hole with a birdie-3 as Weir made par. Harrington went 2-up with a birdie on the par-4 11th and 3-up as Weir bogeyed the 31st hole.
Harrington closed him out on the 33rd hole as Weir made another bogey leaving Harrington with a 4-up victory. Harrington will face world No. 5 Sergio Garcia in the second round on Friday.
Singh, who advanced to the second round last year before being beaten by Sam Torrance, never trailed through the whole match. He made birdie-2 at the par-3 5th and held 3-up leads on two separate occasions but ultimately needed some luck to finally dispose of Rose.
Rose closed the gap to just 1-down as they headed to the 36th hole, the par-5 18th. Singh could only make par and then waited as Rose had a six-footer for birdie to extend the match. An incredulous Rose watched as his putt went halfway down before spinning back out. A relieved Singh goes up against South African Retief Goosen.
Montgomerie used his dominating performance from the Ryder Cup to overwhelm Funk in the days last match. Monty built a 3-up lead during the morning but the feisty Funk got the match to all square at the 20th hole. After halving the next three holes Monty went up for good with a routine par on the par-4 6th.
Monty was 3-up and dormie when they came to the 34th hole. Funks second shot was long and when he didnt hole the chip for birdie he conceded the match to the Scot. Montys victory earned him a shot at Ernie Els in what should be one of the best matches of the tournament.
The fact that Faldos match is yet to be determined is somewhat of a surprise, to both Faldo and Campbell. Down two with two to play, Faldo birdied the par-5 17th and then watched as Campbell missed a four-footer for par at the 36th hole that would have ended the match.
The two then halved the first six holes of sudden death before darkness halted the match. They will complete the round Friday morning with the winner to play last years champion Ian Woosnam.