Exeter were leading 21-3 after just 18 minutes and the hosts looked in grave danger of being routed, but the Exiles rallied to build a 23-21 lead midway through the second half.
It fell to Gareth Steenson to score two late penalties and earn victory for Exeter.
The visitors started brilliantly and took the lead in the third minute with an excellent try. When Irish attempted to relieve the early pressure, the Chiefs counter-attacked through Steenson and Matt Jess to create a try for Phil Dollman.
Steenson missed the conversion but minutes later he would kick a simple penalty.
In the early stages, the visitors were on fire and Steenson made a neat break which was carried on by Richard Baxter and Tom Hayes before Irish were penalised. Steenson made no mistake with the kick and Exeter were 11 points up inside 10 minutes.
Irish responded through an Ian Humphreys penalty, conceded for a high tackle from Hayes, but the hosts made an elementary error straight from the restart to concede a penalty which Steenson kicked.
On almost every occasion that Exeter were in possession, they seemed capable of breaching the home defence and so it came as no surprise when they extended their lead in the 18th minute.
A clever pass from Haydn Thomas sent Jess through another gap and Luke Arscott went over for the try, with Steenson converting.
A deliberate knock on from Dean Mumm conceded a penalty to London Irish which enabled Humphreys to make it 21-6 at the end of the first quarter.
The game continued at a frenetic pace but Exeter became careless with their handling and they soon allowed the hosts to assume an attacking position.
Irish took full advantage with Leo Halavatau driving over from a line-out, leaving Humphreys to convert superbly from the touchline.
It was 21-13 at the interval but Exeter should have been further ahead and they were made to regret it when a clever crossfield kick from Humphreys allowed Topsy Ojo to race in for a try, which Humphreys converted.
Arscott was then penalised for not releasing, allowing Humphreys to hammer over a penalty from inside his own half to put Irish into the lead for the first time with 30 minutes remaining.
Exeter won another penalty for Steenson but the Chiefs received a let-off when Tom Homer missed a penalty from 30 metres, as the full-back took over kicking duties in the temporary absence of the injured Humphreys.
Humphreys soon returned, but Homer continued in the kicking role and came up just short with an attempt from 60 metres.
With 15 remaining, Tom Johnson was introduced in attempt to stem the tide as Irish continued to dominate, but the hosts could not complete what would have been a stunning comeback.
Exeter nearly finished with a flourish against the run of play, but Jack Nowell lost the ball in the process of touching down, before Steenson's fifth penalty of the game sealed the points with one minute remaining.
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