The striker, on loan from Aberdeen, latched onto Scott Fraser's pass for the opener and grabbed his second with a high-class solo run and smart finish.
He dispatched a penalty for his treble by the 30th minute and United could have had several more by half-time.
Sean Welsh's penalty gave Thistle hope but United saw out the win comfortably.
The hosts were sluggish in the first half of their Championship opener against Queen of the South on Saturday but there was nothing to fault the way they started this tie.
United had too much pace and ingenuity for their top-flight opponents,  the outstanding Scott Fraser and Tope Obadeyi causing havoc with their movement.
Obadeyi, twice, and Blair Spittal might have added further goals by the interval but United boss Ray McKinnon manager must have been happy with the three they did score.
Smith showed the benefit of a regular start on loan, taking his goal tally to four for the Terrors, the same number he scored in nearly 90 appearances for Aberdeen, though a high percentage of those outings were as a substitute.
His first was a sublime finish, gathering a delicious Fraser pass, controlling the ball then steering it behind Ryan Scully.
For his second, Smith picked up possession just inside the Thistle half, run away from Callum Booth then held off the defender to score when he got back to him in the box.
The 20-year-old completed his hat trick with a penalty struck straight down the middle after Ziggy Gordon blocked Blair Spittal's shot with an arm.
Thistle were given a glimpse of a way back from a penalty of their own when Chris Erskine was pushed over by Mark Durnan and Welsh scored with 25 minutes left, but United's defence held firm.
United boss Ray McKinnon: "We played some very good football in the first half, especially the forward play, and when the penalty was given against us in the second, I was really pleased with how we reacted.  We showed some resilience.
"Some of the link-up play and finishing was pretty special. Cammy's composure and movement for the goals was outstanding, he has done really well since coming in on loan. I am delighted for him.
"There were huge positives all over. Young left-back Jamie Robson was outstanding, Scott Fraser was outstanding, Blair Spittal was excellent and Tope Obadeyi playing as a lone striker was fantastic."
United hat-trick hero Cammy Smith: "The standard has been set after that. We looked dangerous every time we went forward and could have had a few more goals, but we have to be happy with our night's work.  It is a great victory for us.
"It wasn't like [Aberdeen boss Derek McInnes] was pushing me out the door. We had a few chats and thought going out on loan would be best for my development. I am here to play games and long may it continue."
Thistle boss Alan Archibald: "It was awful right from kick-off - a really poor performance.  I didn't see that coming.
"It was the total opposite of the weekend win [2-0] against Caley Thistle when our keeper didn't have a serious save to make. We were unrecognisable and I feel sorry for our fans that have paid good money to travel here.
"We stood off United from the kick off, made poor decisions right across the pitch and guys who are usually our leaders didn't do it tonight. Nobody got pass marks across the board. Everyone including myself has to look themselves in the mirror."
Match ends, Dundee United 3, Partick Thistle 1.
Second Half ends, Dundee United 3, Partick Thistle 1.
Coll Donaldson (Dundee United) wins a free kick in the attacking half.
Foul by Adebayo Azeez (Partick Thistle).
Foul by Stewart Murdoch (Dundee United).
Adebayo Azeez (Partick Thistle) wins a free kick in the defensive half.
Attempt missed. Blair Spittal (Dundee United) left footed shot from the centre of the box is close, but misses to the right.
Coll Donaldson (Dundee United) wins a free kick in the defensive half.
Foul by Mathias Pogba (Partick Thistle).
Ryan Edwards (Partick Thistle) is shown the yellow card.
Mark Durnan (Dundee United) wins a free kick on the right wing.
Foul by Ryan Edwards (Partick Thistle).
Substitution, Dundee United. Charlie Telfer replaces Scott Fraser because of an injury.
Attempt missed. Ryan Edwards (Partick Thistle) header from the centre of the box misses to the left.
Attempt saved. Simon Murray (Dundee United) right footed shot from outside the box is saved in the bottom left corner.
Foul by Scott Fraser (Dundee United).
Steven Lawless (Partick Thistle) wins a free kick in the defensive half.
Attempt missed. Willo Flood (Dundee United) right footed shot from outside the box is too high.
Substitution, Dundee United. Simon Murray replaces Cameron Smith.
Substitution, Partick Thistle. Mathias Pogba replaces Kris Doolan.
Corner,  Dundee United. Conceded by Ziggy Gordon.
Corner,  Partick Thistle. Conceded by Jamie Robson.
Substitution, Dundee United. Sean Dillon replaces Lewis Toshney because of an injury.
Foul by Coll Donaldson (Dundee United).
(Partick Thistle) wins a free kick in the attacking half.
Callum Booth (Partick Thistle) is shown the yellow card for a bad foul.
Tope Obadeyi (Dundee United) wins a free kick on the right wing.
Foul by Abdul Osman (Partick Thistle).
Willo Flood (Dundee United) wins a free kick in the defensive half.
Foul by Ryan Edwards (Partick Thistle).
Attempt saved. Kris Doolan (Partick Thistle) right footed shot from outside the box is saved in the bottom right corner.
Danny Devine (Partick Thistle) is shown the yellow card for a bad foul.
Scott Fraser (Dundee United) wins a free kick in the defensive half.
Foul by Danny Devine (Partick Thistle).
Goal!  Dundee United 3, Partick Thistle 1. Sean Welsh (Partick Thistle) converts the penalty with a right footed shot to the centre of the goal.
Coll Donaldson (Dundee United) is shown the yellow card.
Penalty conceded by Lewis Toshney (Dundee United) after a foul in the penalty area.
Penalty Partick Thistle. Chris Erskine draws a foul in the penalty area.
Willo Flood (Dundee United) is shown the yellow card for a bad foul.
Foul by Willo Flood (Dundee United).
Cammy Smith's hat-trick inside 20 first-half minutes saw Championship side Dundee United beat Partick Thistle to reach the League Cup quarter-finals.