Wales, who had Neil Taylor sent off for an ugly second-half challenge on Ireland skipper Seamus Coleman, enjoyed the better of the game before the red card but were left hanging on as the Republic launched a late blitz, although Bale went agonisingly to snatching victory.
In the end, a 0-0 draw meant that the spoils were shared as Ireland maintained their record of not losing a home qualifier since September 2013 and Wales were able to console themselves by extending their run to just one defeat in 17 qualification matches.
Serbia edged ahead of the Republic at the top of Group D on goal difference as a result of their 3-1 win in Georgia, with Wales four points adrift alongside Austria at the halfway stage having won just one of their five games to date.
Ireland boss Martin O'Neill, who had seen his squad severely depleted by injury and suspension in the run-up to the game, was dealt a fresh blow minutes before kick-off when James McCarthy, named in the team following his apparent recovery from a hamstring problem, was withdrawn and replaced by David Meyler.
Ominously, however, opposite number Chris Coleman was able to call upon the 11 men who started the famous Euro 2016 quarter-final victory over Belgium for the first time since last summer.
Robbed of the creativity usually supplied by the suspended Robbie Brady and the injured Wes Hoolahan, the Republic were prosaic in approach with wide men Jonathan Walters and James McClean - the latter wearing the number five in memory of close friend and Derry City skipper Ryan McBride - asked to protect their full-backs.
McClean in particular rattled into tackles, the first of them to welcome Bale to Dublin, in a bid to upset the impressive rhythm with which Wales started.
Where the Republic were all sweat and brawn, Chris Coleman's men were fluid and fluent with midfield trio Joe Allen, Aaron Ramsey and Joe Ledley moving the ball confidently to feed wing-backs Chris Gunter and Taylor and try to bring front two Bale and Hal Robson-Kanu into play.
But by the time referee Nicola Rizzoli brought an end to the first half, two markedly methods had yielded much the same - largely nothing.
Glenn Whelan and Meyler both shot well wide from distance and Welsh defender Ben Davies might have been fortunate to escape after McClean's 35th-minute corner appeared to hit his arm inside the box as he attempted to clear, but that was as good as it got for the home side.
For all their possession, Wales fared little better despite getting in behind full-backs Seamus Coleman and Stephen Ward on several occasions, and Bale's wayward attempt from 35 yards five minutes before the break rather summed up their frustration.
It was Bale who produced the game's first attempt on target three minutes after the restart, although his free-kick, which had been awarded for McClean's foul on Allen, was fielded without incident by Darren Randolph.
However, the keeper was relieved to see a blistering left-foot strike by the same man fly a foot wide of his left post seconds later after he took aim from 25 yards.
With half-time substitute Sam Vokes making his presence felt, Ramsey tested Randolph once again with a skidding attempt from distance, but it was Ireland defender Richard Keogh who might finally have opened the scoring on the hour when he met Whelan's corner but bundled it wide.
Lone striker Shane Long scuffed a 62nd-minute volley wide as belatedly Ireland started to make their presence felt.
But the game changed with 21 minutes remaining when, seconds after Bale had been booked for a late challenge on John O'Shea, Taylor was dismissed for an awful challenge which saw Seamus Coleman carried from the field on a stretcher.
McClean came desperately close to winning it for the home side five minutes later when his drilled effort was deflected past the post, but Bale went even closer with a curling effort as time ran down.