The Blues entered the contest off the back of a 3-1 loss at West Ham United, while Luton had not been in action since suffering a 4-1 defeat to Brighton & Hove Albion on the opening weekend of the campaign.
Sterling scored once in either half, while Jackson netted his first goal for his new club late on, with Mauricio Pochettino securing his first victory since arriving as Chelsea head coach over the summer.
The result has left the Blues in eighth position in the table on four points, while Luton are bottom, yet to register a point in England's top flight.
Chelsea's first half-chance of the match arrived in the fifth minute when smart work from Jackson eventually led to Sterling striking towards goal, but Tom Lockyer was on hand to make an excellent block.
Sterling then saw a volley kept out by Luton goalkeeper Thomas Kaminski in the seventh minute, before Tahith Chong fired wide of the post down the other end during a lively start.
Enzo Fernandez was next to try his luck for Chelsea, lifting one over the crossbar in the 15th minute, but the home side managed to make the breakthrough in the 17th minute through Sterling.
It was an excellent solo goal from the England international, who weaved his way into the Luton box from a wide area before picking out the bottom corner with an excellent finish.
Chelsea were the team in control in the latter stages of the first period, but Luton were having their moments on the break, with the promoted outfit still very much in the contest.
The Blues should have doubled their lead in the 50th minute when an excellent team move ended up with Ben Chilwell finding space in a dangerous area, but the left-back decided to look for Sterling when he had a clear sight of goal, as Luton survived conceding a second.
Kaminski then did well to turn a strike from Jackson behind, before Chelsea had a penalty shout when Lockyer pulled Thiago Silva to the ground inside the box, but it was waved away by the referee and VAR agreed.
Fernandez then struck the Luton post just before the hour, with the visitors finding it difficult to stem the flow of traffic at Stamford Bridge.
Chelsea goalkeeper Robert Sanchez had to keep out a strike from Ryan Giles on the hour after smart work from Chong, before Sterling turned one wide of the target down the other end in the 66th minute after more good work from Jackson.
Chelsea managed to double their advantage in the 68th minute, though, with Sterling sweeping the ball into the back of the net from inside the box after meeting a cross from the impressive Malo Gusto.
Fernandez sent a free kick just wide of the Luton post in the 72nd minute, with the hosts chasing a third, and it arrived in the 75th minute, with Jackson converting a Sterling cross from close range.
Kaminski kept out a strike from Conor Gallagher in the 84th minute, and Chelsea ultimately declared at three on a successful night for Pochettino's side.
Chelsea will now switch their attention to the EFL Cup, welcoming AFC Wimbledon to Stamford Bridge in the competition on Wednesday night, while Luton are also in action in the League Cup against Gillingham on Tuesday.