The 32-year-old replaced Billy Vunipola in the Lions squad, after the number eight withdrew with a shoulder injury.
"I am over the moon, but so much can happen," said Haskell.
"Until I'm on the plane I won't really start that enjoyment and celebration."
Wasps face Exeter at Twickenham on Saturday, with the Lions' first match of the New Zealand tour on 3 June.
Haskell told the BBC Rugby Union Weekly podcast: "I was out for lunch with my mum and dad and girlfriend, and I came back in the car and had a missed call from Gats [Warren Gatland].
"He told me the sad news about Billy and asked if I would like to come on tour. I said 'yes sir, whatever you need', and that was it really."
Haskell said Vunipola is "probably the best number eight in the world" and a "huge loss" to head coach Warren Gatland's squad.
"Obviously I'm pleased [to be called up], but as a fan of Billy's and a good mate of his, it's very sad not to see him on the tour," he added.
Despite winning 75 caps over a 10-year England career, Haskell has yet to play for the Lions, missing out on selection in 2009 and 2013.
They play three Tests against the All Blacks, who are reigning world champions, and are aiming to become only the second Lions side to win a series in New Zealand.
"It's the pinnacle of your career and the single best thing you can do in a rugby shirt," said Haskell.
"But I haven't got too excited because we have such a huge game this weekend and anything can happen in rugby.
"I don't want to think about anything else, because I want to give 100% and put my body on the line, and you have to do that thinking there potentially is no tomorrow after the weekend."
Haskell is the sole survivor from Wasps' title-winning side of 2008, and says this current team are determined to create their own legacy against Exeter on Saturday.
Wasps scored a late try to beat Leicester in their semi-final, while Chiefs did the same to see off Saracens.
"It's not about turning up and collecting your pay-check, it's about winning," he said.
"Winning things is what really makes your career. If you don't get the results, it's not worth it really.
"I might cry [if Wasps win]. To lift the Premiership trophy would be a dream."
England flanker James Haskell says his excitement at being called up by the British and Irish Lions has been tempered by his desperation to put his body "on the line" for Wasps in Saturday's Premiership final.