The defeat leaves the Baggies rock bottom of the Premier League with just three matches remaining.
The all-important goal came two minutes before the break. Vedran Corluka cut in from the right and found Jenas who turned and curled a low shot past keeper Scott Carson into the bottom corner.
Spurs shaded the opening 20 minutes. They fashioned a clear chance when Luka Modric was fouled just outside the area but Roman Pavlyuchenko dragged the free-kick wide of the goal.
Albion, who gave themselves a lifeline with a terrific win over Sunderland last week, gradually got into the game, forcing a succession of corners.
And the Baggies came closest to scoring from a flag kick after 22 minutes. Chris Brunt's corner was met by Jonas Olsson but his firm header was superbly saved by keeper Heurelho Gomes.
At the other end Carson stole a low cross from Jenas but Aaron Lennon fired the rebound narrowly wide.
Albion regrouped and Marc-Antoine Fortune seemed to have been bowled over in the area but referee Rob Styles waved play on.
Shortly afterwards Fortune hooked high over the bar but a minute later Spurs broke the deadlock through Jenas.
The Baggies started the second half brightly and Fortune forced Gomes into a low diving save one minute after the restart.
A minute later Albion came within inches of an equaliser when Fortune's deflected drive beat Gomes but cannoned off the inside of the post and the rebound eluded the inrushing Brunt.
Spurs gradually forced their way back into the game and Lennon burst clear from midfield forcing Carson into a save at the second attempt.
The speedy Spurs winger then found space on the right, cut into the box and his low cross just eluded the inrushing Robbie Keane.
The Baggies lifted the tempo and Shelton Martis headed over from Brunt's free-kick.
West Brom went even closer a minute later when Martis' header deflected off Jermain Defoe on the line and Gomes pulled off a brilliant reaction stop to keep the ball out.
The Brazilian keeper again came to Spurs' rescue when he palmed out a deflected shot from Robert Koren.
The game opened up in the closing minutes as West Brom chased an equaliser. But it was Spurs who came closest to scoring, Lennon hitting the inside of a post after being released by Defoe.
And in stoppage time Jenas forced Carson to beat out a low drive out for a corner.