After Lloyd Young had given Montrose an early lead, the struggling hosts went ahead as Kevin Turner took his goals tally for the season to nine with a quick brace.
However, Wood's equaliser ensured Montrose moved above Peterhead into fourth place in the table and extended their unbeaten run on the road to four matches.
Montrose went ahead in the 12th minute when Young found the top corner from 12 yards.
Leighton McIntosh could have doubled Montrose's advantage, only to miss from close range, before East Stirling punished them with a leveller in the 24th minute.
After taking on a pass from Jamie McKernon, Turner struck the ball into the bottom corner from 12 yards and then converted from close range on the stroke of half-time.
Montrose were back on level terms 18 minutes after the interval as Wood directed Scott Johnston's cross into the net.
Turner almost grabbed his hat-trick 10 minutes from time but he shot wide from distance.