Cleverley’s close-range finish just after the half-hour mark thanks to fine play by Jack Grealish gave Villa the three points, as the hosts came through a nervy second half that lacked chances for both sides.
The result leaves Villa four points above the drop zone with two games remaining, and marked their first back-to-back home wins of the season.
Tim Sherwood named an unchanged side from the team that beat Everton 3-2, but were under pressure before a ball was kicked after Sunderland also overcame the Toffees 2-0 in the early kick-off to leave Villa just a point above the drop zone.
The hosts started the contest the brighter, and went close after six minutes through Leandro Bacuna’s curling free-kick.
Charles N’Zogbia’s goal-bound effort from 12 yards was blocked well by Kevin Nolan, before Adrian parried a Fabian Delph effort away from danger.
Christian Benteke failed to connect well enough on the turn from a Jores Okore’s header, while Cleverley forced another save out of Adrian with a volley from the right corner of the penalty area.
But the on-loan Manchester United midfielder earned his third goal in three games on 31 minutes, largely thanks to youngster Grealish’s fine wing play.
Collecting N’Zogbia’s pass, the 19-year-old barged his way to the byline, before finding Cleverley with a fine cut-back for the midfielder to turn home from three yards.
West Ham brought Nene and Alex Song on at half-time, and the former’s effort from the edge of the box fizzed just over Given’s bar a minute after the restart.
Given then tipped over James Collins’ header from a corner, as the prospect of a third win in four games created a nervous atmosphere at Villa Park.
Despite late pressure from the visitors, it was Villa who came closest to scoring the second, but substitute Gabriel Agbonlahor made a hash of his effort after being put through on goal by Grealish.
After Hull’s defeat by relegated Burnley on Saturday, Villa can finally see some daylight between themselves and the bottom three having picked up 10 points out of a possible 15.
Aston Villa: Given (7), Bacuna (6), Okore (7), Vlaar (6), Richardson (6), Westwood (7), Cleverley (8), Delph (8), N’Zogbia (7), Grealish (8), Benteke (6)
Subs:, Agbonlahor (5), Hutton (5), Sanchez (5)
West Ham: Adrian (6), Jenkinson (5), Burke (5), Collins (6), Cresswell (6), Noble (5), Kouyate (5), Downing (5), Nolan (5), Amalfitano (5), Valencia (5)
Subs:, Nene (6), Cole (5), Song (5)
Man of the match: Tom Cleverley