The result means West Brom are almost certainly going to be relegated from the Premier League in a game which they will be disappointed to have lost.
There were chances at either end before Jenas opened the scoring on 43 minutes, coolly stroking the ball home from just outside the box to give Spurs a half-time lead.
West Brom boss Tony Mowbray made several changes in the second half and his side may live to regret missed opportunities, most notably from Shelton Martis who squandered a free header while Marc-Antoine Fortune hit the post with a deflected effort.
The three points puts Tottenham back in the frame for a place in next season's Europa League as they move level on points with Fulham and Man City, albeit with an inferior goal difference.
Meanwhile the Baggies have to host Liverpool and Wigan before travelling to Blackburn and seven points could still be enough for them, but time appears to be running out.
The first half summed up their season, they had plenty of possession, created chances but found themselves going into the break behind.
ChancesThey would have been ahead after 22 minutes but Spurs goalkeeper Heurelho Gomes kept the scores level with an acrobatic save, with Jonas Olsson almost turning around to celebrate before the Brazilian plucked his header away.
The Baggies had a penalty shout turned down when Benoit Assou-Ekotto got his body in front of 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, with Aaron Lennon stabbing wide after Scott Carson fumbled and Roman Pavlyuchenko thrashing over after Luka Modric slid him through down the left.
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.
HarshIt was harsh on West Brom as Carson looked to have had his view blocked by referee Styles but that mattered little to Jenas, who would have been aware Fabio Capello was watching in the stands.
Capello was keeping an eye on the likes of Lennon, as well as Croatia pair Vedran Corluka and Modric, who England will be up against later this year in a World Cup qualifier.
Spurs' England centre-backs, Ledley King and Jonathan Woodgate, were trying their best to keep Fortune quiet. West Brom's lone forward forced Gomes into a save with a low shot early in the second half before stricking the post with an effort that deflected off Corluka.
Mowbray then brought on Jay Simpson to offer Fortune support, while Jermain Defoe was introduced for Pavlyuchenko.
Martis had one header flash over and another cleared off the line by Defoe, while Borja Valero cleared his own crossbar when he made a last-ditch clearance at the other end.
Gomes was forced into a late save, at full stretch, to deny Robert Koren's deflected effort before Lennon struck a post in the final minute.
West Bromwich Albion
1st Half Goals
Shots on Target
Shots off Target