The Chiefs trailed 26-19 with less than two minutes remaining before tries from Phil Dollman and Sireli Naqelevuki and a conversion from Gareth Steenson broke the Warriors' hearts.
And Baxter said: "The boys are elated. It's a fantastic achievement to qualify for the Heineken. We also have Northampton at home next Sunday in what is a virtual winner-takes-all game for a Premiership play-off spot."
Baxter accepted his side's good fortune in winning a thrilling match courtesy of their dramatic late rally.
"I thought it was a loss, we weren't quite on our game today but it takes a good side to score four tries against such a tight defence as Worcester's," he said. "We gave away too many penalties."
Exeter's discipline was indeed poor as they allowed Andy Goode to kick four penalties, with the Chiefs also having two players sent to the sinbin including Gonzalo Camacho, who was somewhat fortunate not to pick up a red card.
Baxter said: "It looked bad but the referee thought about it and came up with the right decision. I didn't think it was a spear tackle but it was borderline as he (Alex Grove) was flipped over."
Tries from Luke Arscott and Bryan Rennie put Exeter 16-14 up at half-time, Craig Gillies and Miles Benjamin with Worcester's replies.
Goode kicked four second half penalties for the Warriors but they failed to make Exeter pay a steeper price for their indiscipline, and it backfired in the closing stages.
Worcester coach Richard Hill was downcast after his side's defeat.
"It was a massive blow to us," he said.
"It's not very pleasant, we should have kicked long with the final restart to give them more to do.
"We blew a couple of opportunities and should have made more of their two players in the sinbin but I still thought that we were in control of the game."
Worcester still face the lingering threat of relegation but Hill is confident they will remain in the top flight.
"We are still not mathematically safe but it will be difficult for Newcastle to pick up two bonus point wins against Saracens and Wasps. However, we have to make sure we win at home to London Irish next week."
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