Joe Fitzpatrick gave the visitors the lead after 38 minutes before Shifnal's Tyrone Ofori equalised just before half-time.
Only four minutes after the break Matthew Hughes restored Wood's lead before an instant response from Shifnal, inside a minute, saw Michael Nelson draw them level.
Wood went ahead for the third time in the match when Adam McGurk struck in the 68th minute.
The visitors looked set to take all three points until the hosts equalised for a third time, in the 91st minute, through Thomas Hill.
Shifnal are fifth in the division, level on points with fourth-placed Darlaston Town, who they have a game in hand over.
The Shropshire outfit also have two games in hand on all of the top three teams and only a three-point gap stands in the way of them catching second-placed Stourport Swifts.
Elsewhere in the league, Market Drayton Town crashed to their 14th consecutive league defeat, as they lost 1-0 away at Highgate United. A 57th minute goal from Jack Till separated the sides.
Drayton remain rooted at the bottom of the league with one win from 15 games, which came on their opening league fixture back in August.
Meanwhile, the game between Tividale and Whitchurch Alport was abandoned at half-time due to an alleged racist comment directed at a Tividale player from a member of the crowd.
In Midland League Division One, AFC Bridgnorth lost 3-1 in a crunch clash away to Coventry Copsewood.
With both clubs languishing towards the wrong end of the table, Bridgnorth failed to build from their win last week with a poor defeat.
They sit 16th in the league, now just two points ahead of 18th-placed Copsewood and just four points off the relegation zone.
Haughmond also tasted defeat, losing 3-1 to Wednesfield, while Shawbury United were beaten 2-0 by Stapenhill.
In the North West Counties League Division One South, Ellesmere Rangers suffered a blow in their promotion race as they lost 3-1 to strugglers Cheadle Heath Nomads.
The visitors raced into a 3-0 lead and Steven Hole's goal for Ellesmere was nothing more than a consolation.
After the defeat, Ellesmere sit third in the table and nine points off leaders Stockport Town, while the winners on the day are in the relegation zone.