Some ferocious tackling by Connacht's Ray Ofisa, Michael Swift and George Naoupu held back the Exeter attack. By the quarter hour mark, the visitors had not added to their tally despite having the majority of possession.
Referee Dudley Phillips kept a firm grip on the scrums and penalised Connacht on a number of occasions for technical infringements.
This gave Exeter both possession and points as Steenson extended their lead to 6-0 before Connacht managed to hit back through new out-half Niall O'Connor.
A push on his half-back partner O'Donohoe allowed O'Connor convert a 35-metre penalty, but it was the Chiefs who scored the first try in the 29th minute.
Centre Bryan Rennie took a pop pass from full-back Luke Arscott to score from close range, with Steenson adding the conversion.
The Dungannon-born number 10 tagged on a penalty after Connacht were deemed offside, a decision questioned by many in the stands and he then fired over another successful kick following a scrum penalty.
Steenson was on hand at just the right moment to intercept an inside pass from Fa'afili which was meant for Brian Tuohy, and sprint from the halfway line to score under the posts.
He added the conversion on the stroke of half-time to give Exeter a 26-3 lead though some strong defensive work from Connacht did not do justice to the scoreline.
An early series of penalties in the second half let Connacht pile on the pressure and work their way up towards the Exeter try-line.
Eric Elwood's men threatened with a lineout drive and then a cross-field attack, but referee Phillips blew for hands in the ruck and the Chiefs were able to clear their lines.
O'Connor's boot kept Connacht in the Exeter half and the hosts were now enjoying the majority of possession. However, a long stoppage in play for an injury to lock Dave Nolan unfortunately broke up their momentum.
Nolan was tackled in the air at a lineout 15 metres out and had to be stretchered off the pitch.
When play resumed, Exeter seized the initiative again. Replacement centre Sireli Naqelevuki made a break through the middle and combined with Jason Shoemark, who passed for the onrushing Phil Dollman to score close to the posts.
Steenson added the extras and Exeter cut through for their fourth try, just three minutes later. Kevin Barrett took a quick tap penalty and he found fellow replacement Nic Sestaret who found a hole in the Connacht defence and darted through from 25 metres out.
Steenson's successful conversion made it 40-3 to the Aviva Premiership outfit. Despite the one-sided nature of the game, there was still plenty of spice to the exchanges and when things bubbled over, hookers Ethienne Reynecke and Neil Clark were promptly sin-binned.
A fifth try followed for Exeter's Shoemark, with Steenson's replacement Ignacio Mieres converting, before newcomer Matthew Jarvis notched a late consolation try for Connacht.
In injury-time, Jarvis picked off a pass from Junior Poluleuligaga on the Exeter 22-metre line to cross the whitewash. He kicked the conversion himself as the westerners finished the game on a high note..
CONNACHT: Tiernan O'Halloran; Mark McCrea, Eoin Griffin, Henry Fa'afili, Brian Tuohy; Niall O'Connor, Paul O'Donohoe; Dylan Rogers, Ethienne Reynecke, Rodney Ah You, Michael Swift, Dave Nolan, Thomas Anderson, Ray Ofisa, George Naoupu.
Replacements (all used): Adrian Flavin, Denis Buckley, Michael Kearney, John Muldoon, Eoin McKeon, Dave Moore, Matthew Jarvis, James Loxton, Dave McSharry.
EXETER CHIEFS: Luke Arscott; Josh Tatupu, Jason Shoemark, Bryan Rennie, Matt Jess; Gareth Steenson, Haydn Thomas; Brett Sturgess, Chris Whitehead, Hoani Tui, James Hanks, Peter Short, Tom Johnson, James Scaysbrook, Richard Baxter.
Replacements (all used): Phil Dollman, Ben Moon, Neil Clark, Nic Sestaret, Sireli Naqelevuki, Mark Foster, Ignacio Mieres, Kevin Barrett, Junior Poluleuligaga, Simon Alcott, Chris Budgen, John Andress, Aly Muldowney, Chris Bentley, Chad Slade, Ben White, James Phillips.
Referee: Dudley Phillips (IRFU)