Sunday's Madejski Stadium triumph was the Chiefs' first away win of the Aviva Premiership season.
Exeter led 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.
However, the visitors regrouped and it fell to Gareth Steenson to score two late penalties and earn a 27-23 victory for Exeter.
Baxter said: "For the first 20 minutes, we were very good and very dangerous as we were looking very direct and controlling the game.
"But then we started playing from side to side, just making passes and leaving the final man isolated. As a result we lost huge field positions.
"We made hard work of it and it's certainly a relief to pick up four points on the road. We've never won here and haven't won away in the Premiership so it's great to get the monkey off our back.
"They were without a number of players so on paper it's a game we should have won,
but they were very tough and dogged in defence and in the end we had to grind it out."
London Irish director of rugby Brian Smith said: "They got a flyer in the first 20 minutes, we lost concentration and we lost our focus in defence.
"We gave ourselves a mountain to climb but with our try just before half-time it gave us a big momentum shift.
"Because they had such a good start it was a bit of a negative for them as they stopped playing.
"Our improved performance gave us a shot to win and we did a good job of running them down but they nailed their penalties at the crucial time and we missed ours.
"We had eight players away on international duty and another eight injured so it was good to salvage a bonus point."
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