Paqueta looked to have sealed West Ham's second successive Premier League victory when he converted from the spot with just three minutes left at the Vitality Stadium.
But Unal came to the Cherries' rescue three minutes later with only his second goal this season.
The 27-year-old's moment of magic extended Bournemouth's unbeaten run to four games as they climbed to sixth place.
Bournemouth have scored a league-high 10 goals in the final 15 minutes of games this season.
Two of those goals came in a remarkable 2-1 win at Ipswich in their previous game, with Unal also scoring Bournemouth's equaliser on that occasion.
West Ham have gone two games without defeat to ease a little of the pressure on boss Julen Lopetegui.
They remain in 14th place and sit seven points clear of the relegation zone.
Bournemouth have beaten Tottenham, Manchester City and Arsenal at home this season, so the under-fire Lopetegui might have feared the worst when Andoni Iraola's team made a strong start.
Antoine Semenyo was the catalyst for Bournemouth's first threat as he won a towering header before sprinting into the area to meet Evanilson's return pass with a low strike that glanced off the far post.
West Ham forward Jarrod Bowen scored the late winner when the sides last met in the League Cup in August and he nearly tormented Bournemouth again with a curler from the edge of the area that clanged off the crossbar.
Bowen went close again with a snap-shot from Aaron Wan-Bissaka's cross, but Kepa Arrizabalaga was equal to the effort.
Lopetegui's men were a threat on the counter and Carlos Soler whipped a shot narrowly wide before Tomas Soucek dragged a close-range effort just past the post.
Bournemouth's Dango Ouattara looked certain to break the deadlock when the ball fell to him just five yards from goal, but Lukasz Fabianski threw himself across to make a superb save with his legs.
Ouattara almost bent a long-range effort past Fabianski soon after the interval, while Ryan Christie's drive was tipped over by the Hammers 'keeper.
The chances kept coming for Bournemouth as Illia Zabarnyi's header whistled wide, then Fabianski saved Evanilson's glancing header.
After Fabianski denied Philip Billing and pushed away Christie's follow-up, West Ham took the lead in the 87th minute.
Tyler Adams blocked Wan-Bissaka's cross with his out-stretched arm and referee Chris Kavanagh gave the penalty after consulting the pitchside monitor.
Paqueta took the spot-kick and calmly sent Kepa the wrong way with his stutter-step run-up.
But there was a sting in the tale in the 90th minute as Unal bagged a brilliant equaliser, looping an unstoppable free-kick into the top corner from 25 yards.
© 2024 AFP