Minutes after James McClean's own-goal from a deflected Alexis Sánchez free-kick had looked set to inflict more festive misery on Baggies boss Alan Pardew, Rodriguez coolly stepped up after Calum Chambers was adjudged to have handled in the penalty area in a dramatic conclusion to the calendar year.
The point still leaves the hosts in the bottom three going into 2018, as Pardew remains winless in his new job.
Baggies keep visitors at bay
With the visitors making just the one enforced change with Alex Iwobi coming in for the absent Mesut Özil, Arsène Wenger's men kept to the same wing-back system that saw Arsenal take the spoils at Selhurst Park earlier in the week, but in the early going were finding life difficult in terms of creating meaningful chances.
West Brom themselves were finding the width in the wide areas from both Jay Rodriguez and Matt Phillips, and were causing the Gunners problems. The duo combined in the early going, with the Scottish international's looping delivery finding Rodriguez at the far post, but whose header goal-wards was easily held by Petr Cech.
The Arsenal threat had been largely snuffed out in the opening period except for two chances from outside the box just before the half-hour mark, with both Granit Xhaka and Iwobi's efforts respectively, dealt with by Baggies' 'keeper Ben Foster.
Whilst the hosts had retreated into their shell as the half wore on, the space was still being afforded them in attack, and as the half-time whistle blew Alan Pardew's set-up at least, was paying off.
Patient approach continues
Both sides were in need of a dash of New Year vigour on resumption in the West Midlands and it was West Brom who again showed the greater energy.
It was no surprise to see the wide approach continuing from Pardew's men and it was Ahmed Hegazi who was the recipient of a Rodriguez knock-down from a set-piece in the 56th-minute, but after the Egyptian failed to control a lively ball in the area, the defender's acrobatic half-volley effort was skewed wayward.
Alexandre Lacazette was finally making his presence felt however for Arsenal, and twice went close to breaking the deadlock, but was denied in the first instance by the legs of Foster, before the former Lyon man saw his touch go inches past the near post just after the hour mark.
Frenetic end at Hawthorns
As the game entered its' final 15 minutes both sides were looking to snatch the points. Iwobi again shied from distance, before Rodriguez cracked a shot straight at Cech thirteen minutes from time, with substitute McClean unable to reach the rebound quick enough before being cleared.
The Irishman was however then involved at the other end from - rather ironically - a free-kick, in what was a frenetic end to proceedings. From the ensuing kick, Sanchez stepped up with his shot hitting the wall only for the Chilean's effort - seemingly bending wide of the target - to hit the back of McClean and deflect under Foster and into the net. It triggered an immediate response from the hosts under controversial circumstances.
As a hopeful ball found former Gunner Kieran Gibbs on the edge of the box, his flicked effort to keep the ball in hit Chambers on the hand, with Mike Dean ruling a handball.
Rodriguez, who had had a lively 90 minutes for the Baggies, coolly converted the penalty down the middle. It was a dramatic end to what had largely been a frustrating New Years Eve, with both sides forced to settle for a share of the spoils to round off 2017.
Both sides face crucial games next up, with West Brom traveling to West Ham on Tuesday night, whilst Arsenal host Chelsea in a London derby on Wednesday.