Everton moved to within a point of a Champions League place after they beat struggling West Ham at Goodison Park.
Tevez has still to score for West Ham this season
Everton went ahead early in the second half when Leon Osman guided a volley past Robert Green after West Ham failed to clear James Beattie's cross.
In stoppage time James Vaughan drilled a low shot under Green for the second.
West Ham had the better chances in the opening half, notably when Lee Bowyer was thwarted by a Tim Howard save after good work by Carlos Tevez.
After Osman scored, Howard made an important save to beat away substitute Marlon Harewood's powerful shot.
The win moved Everton to within a point of Arsenal and Portsmouth, who are in third and fourth place respectively.
Defeat left West Ham a point ahead of third-from-bottom Newcastle and without a league goal away from home for over three months.
The frustration for West Ham was that in Tevez the visitors probably had the game's most dangerous attacking player.
Tevez has yet to score since arriving at West Ham, but at a dank and windy Goodison his skill and invention constantly illuminated the game and provided encouraging signs he is starting to settle in England.
The Argentine's ability to beat his man provided an opening for Bowyer that really should have given the visitors the lead.
Tevez jinked his way past Joleon Lescott before cutting the ball back to Bowyer, whose sidefooted volley was pushed away by Howard.
Having set up Bowyer on the right, Tevez switched flanks, cutting inside to try his luck with a couple of rising shots.
So in control were West Ham that Everton's first chance did not materialise until just before the half-hour.
Andy Johnson was quick to react after the ball ricocheted off Anton Ferdinand and his skimming shot was pushed away by Green.
The West Ham goalkeeper was soon in action again when he came out to smother the ball at Beattie's feet.
Tevez made a bright start to the second half, delivering a beautifully weighted pass to Bobby Zamora, who shot narrowly over the bar.
Osman had failed to score this season before Sunday, but on 51 minutes he corrected his goalscoring statistics, cleverly guiding the ball past Green.
The goal owed much to Beattie's persistence in the way the Everton striker chased a Yobo pass that had been overhit.
Beattie's diligence enabled him to send over a cross that Jonathan Spector was unable to get any distance on his headed clearance and Osman was quick to take advantage.
The 18-year-old Vaughan also claimed his first goal of the season when his low shot squirmed under Green.
Everton boss David Moyes:
"It's an unbelievable result for us.
"I don't know how they did it. I think we ended up with five centre halves on the pitch and four centre forwards.
"Everyone was playing out of position but I think that epitomises the spirit we have."
West Ham boss Alan Pardew:
"The goal drought is a concern. In terms of penetration and control of the game, up until their goal our performance was very good.
"David Moyes almost apologised to me at the end because he must feel he's got a win from nothing.
"Carlos Tevez needs a goal but his display was a big positive for us."
Everton: Howard, Yobo, Stubbs, Lescott, Nuno Valente, Van der Meyde (Vaughan 72), Osman (Weir 82), Carsley, McFadden, Johnson, Beattie (Hughes 90).
Subs Not Used: Wright, Vidarsson.
Booked: Osman, McFadden.
Goals: Osman 51, Vaughan 90.
West Ham: Green, Spector (Mascherano 84), Ferdinand, Collins, Konchesky, Bowyer (Sheringham 69), Mullins, Reo-Coker, Etherington, Zamora (Harewood 69), Tevez.
Subs Not Used: Kiraly, McCartney.
Booked: Bowyer, Collins.
Ref: M Atkinson (W Yorkshire).