Centres Joe Ansbro and Shontayne Hape grabbed Irish tries, while wing Tom Homer landed 19 points with five penalties and two conversions.
Exeter were always behind but managed a losing bonus point after full-back Luke Arscott got a try in the second half while fly-half Ignacio Mieres kicked five penalties and a conversion.
However, both teams were lucky not to have had a man or two in the sin-bin during the first half as Irish went into the break 19-9 in front.
The Exiles started the match with a bang when Ansbro grabbed their first try inside three minutes, which was more about a superb break from his own half down the pitch by ex-England wing Topsy Ojo.
The flyer saw a gap in the Exeter midfield and took full advantage of it. Exeter covered but Ojo was able to get his pass away through supporting number eight Richard Thorpe for the back row to then feed Ansbro 10 metres out for a great try.
Homer landed the conversion and then swapped a penalty with Mieres as Irish led 10-3.
But that preceded two incidents involving England full-back Delon Armitage for which referee Dean Richards could have awarded a yellow card.
Armitage was victim and then villain. First, he made a terrific break in his own half and kicked downfield. He was racing to a possible score but was impeded off the ball and knocked to the ground when Exeter flanker James Scaysbrook came tumbling in 10 metres from the Chiefs line.
Homer and Mieres continued to swap kicks before the Argentinian stand-off was then the victim of a high tackle from Armitage when fielding an up-and-under.
There was hardly a dull moment as, despite the errors, both teams tried to play some rugby but it was Exeter who were starting to hit back through their pack in particular.
Hooker Simon Alcott got the travelling support on their feet with a dash which was stopped well by the home defence.
Exeter pulled back to 16-9 in arrears and again Irish could feel lucky not to have had another player in the bin as the Chiefs were only three metres from scoring when flanker Jamie Gibson took an age to get his body out of the way, thus preventing the ball appearing.
Referee Richards awarded a penalty under the posts. The Exiles, though, went into the interval 10 points to the good when Homer booted his fourth shot at goal.
But the Chiefs, who fought back from 23-7 down against Sale to win only a few weeks ago, started to turn the screw after the break.
A forward pass from scrum-half Haydn Thomas denied Arscott a try before England centre Hape was sent over after a classic set-piece attack inside the Exeter 22 when the ball was sent through the half-backs for Armitage to put Hape in past some weak tackling.
Yet the Chiefs would not lay down, and 27-year-old Arscott found his way over when Thomas sent a short ball to him at an attacking ruck.
Irish put the clamps on the match, tied Exeter's threat down and, although Mieres landed a final penalty seven minutes from the end, it was not enough as Irish climbed the table.
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