Winger Jennings settled the contest early in the second half with a powerful low drive from the edge of the box to lift Barnsley to 23rd - two points from safety - and stretch Forest's winless run to five games in all competitions.
The visitors, who remain in the play-off positions despite the defeat, came closest to a goal when Djamel Abdoun's first-half strike hit a post.
Jennings threatened early on with a powerful shot from distance which was tipped over the crossbar by visiting goalkeeper Karl Darlow.
Barnsley midfielder Martin Woods then fired wide of the left-hand upright from the edge of the 18-yard box with 17 minutes played.
Forest began to assert some pressure at the other end and almost took the lead six minutes later through wideman Abdoun.
A ball into the Barnsley box came back off Forest winger Jamie Mackie and the Algeria international smashed a low drive which thumped off the base of the right-hand post and away to safety.
Abdoun then turned creator, sending a pinpoint cross from the left into the path of Mackie only to see the former QPR man somehow head wide from inside Luke Steele's six-yard box.
The visitors continued to look the more dangerous going forward and Radoslaw Majewski and young forward Jamie Paterson both had efforts blocked before Polish midfielder Majewski saw his half-volley deflect wide for a corner.
Forest manager Billy Davies, who was forced to watch the game from the stands as he began a five-match touchline ban for abusing a referee, saw his side fall behind six minutes after the interval.
The lively Jennings picked the ball up on the left, just outside the Forest area, before advancing and striking a low right-footed shot into the bottom left-hand corner from 20 yards.
Forest attempted to respond but both Paterson and Abdoun failed to hit the target with shots from range.
Substitute Darius Henderson then wasted a promising opening for the visitors in the 72nd minute when he scuffed his shot into the ground after being picked out in the area by Abdoun, before Paterson nodded straight at Tykes keeper Steele with a free header moments later.
Barnsley became encamped in their own half in the final stages of the game and Steele was forced to produce a fine save from Abdoun's injury-time free-kick as Danny Wilson's men extended their unbeaten home record to five games.