Tevez was lurking to turn in Wayne Rooney's driven shot from a Ryan Giggs corner three minutes before half-time.
Liverpool had earlier come close, but Anderson scrambled Harry Kewell's shot off the line and Fernando Torres sent a header inches wide.
Rooney wasted a great chance to double United's lead late on, but he side-footed wide from only six yards.
It made uncomfortable viewing for Liverpool co-owners Tom Hicks and George Gillett, at Anfield before their summit meeting with boss Rafael Benitez after their recent disagreements.
The defeat leaves the Merseysiders outside the top four and nine points behind arch rivals United.
Liverpool, unchanged for only the third time in Benitez's reign, created the clearer openings in a disjointed first 45 minutes.
Jamie Carragher's 27th-minute header created confusion in the United penalty area, and when Anderson cleared Kewell's shot off the line, Torres headed just wide.
United were under pressure again three minutes later when keeper Edwin van der Sar misjudged Steven Gerrard's free-kick.
The ball bounced towards goal off Rio Ferdinand's head, but once again Evra was alert to clear as Dirk Kuyt closed in.
United capped their best spell of the game by taking the lead three minutes before the interval.
Liverpool boss Benitez was left fuming as his side failed to mark Rooney from Giggs' corner, and when the former Everton star lashed the ball into the danger area, Tevez was on hand to turn home from six yards out.
United were surviving in relative comfort, and Benitez made an inevitable change after 65 minutes, replacing the disappointing Kewell with Ryan Babel, with Kuyt quickly following as Peter Crouch came on.
It at least sparked Liverpool's best effort of the game as Babel shot inches wide from 25 yards after collecting Wes Brown's poor header.
Rooney then wasted a glorious opportunity to seal the points for United at the end of a lightning break, but he shot hopelessly wide from close range from Cristiano Ronaldo's pass.
Liverpool pressed, but lacked any ideas going forward and United's defensive resilience earned the three points.
Liverpool: Reina, Arbeloa, Hyypia, Carragher, Riise (Aurelio 80), Benayoun, Gerrard, Mascherano, Kewell (Babel 65), Torres, Kuyt (Crouch 73). Subs Not Used: Itandje, Lucas.
Booked: Gerrard, Mascherano.
Man Utd: Van der Sar, Brown, Ferdinand, Vidic, Evra, Ronaldo, Hargreaves, Anderson (O'Shea 90), Giggs, Rooney, Tevez (Carrick 83). Subs Not Used: Kuszczak, Saha, Fletcher.
Booked: Evra, Anderson, Brown.
Goals: Tevez 43.
Att: 44,459.
Ref: M Halsey (Lancashire).
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