Jenas, with England boss Fabio Capello in the stands, broke the deadlock just before the break to keep his side in the hunt for a seventh-placed finish that would earn European football.
The bottom-placed visitors had a penalty shout turned down in the first half and Shelton Martis had a header cleared off the line.
They would have been ahead after 22 minutes but Spurs goalkeeper Heurelho Gomes kept the scores level with an acrobatic save from Jonas Olsson's header.
The Baggies had a penalty shout turned down when Benoit Assou-Ekotto got his body in front of Marc-Antoine Fortune, with Rob Styles giving the benefit of the doubt to the Spurs left-back.
Fortune also volleyed over after being found unmarked at the far post by Gianni Zuiverloon's cross. Had one of those chances resulted in a goal, there may have been a different outcome.
Spurs had been limited to half-chances for Aaron Lennon and Roman Pavlyuchenko.
There would have been no complaints if the score was level at half-time but Jenas broke the deadlock two minutes before the break. He received the ball from Vedran Corluka, had time to take a touch as Youssouf Mulumbu backed off, before curling his effort into the bottom corner from 25 yards.
In the second half Mowbray brought on Jay Simpson to offer Fortune support. Martis had one header flash over and another cleared off the line by substitute Jermain Defoe, while Borja Valero cleared his own crossbar when he made a last-ditch clearance.
Gomes was forced into a late save at full stretch to deny Robert Koren's deflected effort and Lennon struck a post in the final minute.