Motherwell had missed a handful of good chances before the unmarked Sutton scored a diving header from an Iain Vigurs cross in a 54th-minute counter-attack.
Kris Boyd missed a great chance to equalise and Anier came off the bench to round off the victory with an opportunist goal with two minutes remaining.
The pair now have 11 league goals between them this season, with Sutton one ahead, and their latest strikes took Stuart McCall's team into third place.
Kilmarnock had won their previous two home games but their third successive defeat left them second bottom and now 12 points above Hearts.
Kilmarnock brought in 19-year-old midfielders Craig Slater and Chris Johnston for the suspended Darren Barr and Jackson Irvine both made lively starts with the latter's direct running causing problems for Motherwell in the early stages.
But it was a teenager in claret and amber who soon took centre stage as Craig Moore missed three chances to mark his first Motherwell start with a goal.
Moore, who scored his first goal for Motherwell against Hearts last month, firstly robbed Mark O'Hara on the edge of the box but his attempt to place the ball in the far top corner was just too high.
The ball then fell for Moore on the penalty spot but he sliced a volley wide. He was soon clean through after O'Hara slipped while trying to cut out Paul Lawson's pass but he had more time than he used as he fired just wide of the far post with his left foot.
Moore was one of four players brought in following Motherwell's 4-0 defeat by Dundee United, with Shaun Hutchinson back from suspension in a defence still missing Simon Ramsden and Steven Hammell.
Motherwell had a more solid look about them but Kilmarnock came close from Sammy Clingan's free-kick and Boyd almost got away from the visiting defence until Stephen McManus recovered.
The move of the half came from Motherwell when Fraser Kerr planted a first-time cross on Sutton's head after a one-two with James McFadden but Craig Samson clawed the striker's central effort over the bar.
Gunnar Nielsen stopped Boyd's ambitious effort before the break and held a long-range strike from Johnston after it.
But Motherwell soon missed another good chance when Lionel Ainsworth, who replaced McFadden at half-time, fired wide from 18 yards after Moore's backheel.
They finally took a chance four minutes later after Zaine Francis-Angol sent Ainsworth away down the right as Motherwell broke following a Killie corner.
The winger's instant cross was too far in front of Vigurs but the former Ross County man swung the ball in from the left side and Sutton had all the time and space he needed to watch the ball bounce and stoop to head into the corner of the net.
Killie almost hit back instantly following an excellent run from Johnston, but Nielsen got down well to stop Michael Gardyne's strike and Boyd could not get a touch when Jeroen Tesselaar drove the rebound across the face of goal.
Sutton and Hutchinson both just failed to convert a corner before Boyd missed a sitter in the 66th minute. Gardyne's cross hung up perfectly for the recalled Scotland striker but he sent a free header over from six yards.
Sutton appealed unsuccessfully for handball as the grounded Manuel Pascali blocked his close-range effort before Motherwell survived a couple of scares when Killie substitute Rory McKenzie fired a long-range effort from distance that bounced just wide and Nielsen held Pascali's header.
But Sutton came close following good link play with Kerr and Anier fired through Samson's legs from eight yards after making a chance for himself with some persistent play.