Leeds striker Becchio headed home his 11th goal of the season in the 73rd minute to cancel out Joe Mason's opener for Cardiff just before half-time and leave the Welsh side with a solitary point from their final home game of the season in the Championship on Saturday.
It was no less than Leeds deserved for a spirited second-half display and Cardiff, unbeaten now in their last nine matches - albeit with six draws - face a nerve-jangling last-day finale at Crystal Palace in their bid to make it a hat-trick of top-six finishes.
Malky Mackay's side had put destiny in their own hands with two wins in the last seven days, but Becchio's header means the Bluebirds will require at least a point next weekend at Selhurst Park if closest challengers Middlesbrough beat Southampton later on Saturday.
Cardiff were unchanged from the side that beat Derby on Tuesday night, while Alex Bruce, Zac Thompson and Becchio returned for Leeds.
Bluebirds midfielder Liam Lawrence had an early effort saved by Leeds goalkeeper Andy Lonergan, and at the other end Ross McCormack headed wide against his former club after good work from Robert Snodgrass.
Mason skewed his close-range effort off target as Cardiff sought an early breakthrough, but neither side created a further chance before Leeds defender Leigh Bromby was stretchered off in the 32nd minute.
Bromby, who appeared to slip awkwardly unopposed and required immediate medical attention, was replaced by Paul Connolly.
That setback appeared to distract the visitors and Cardiff caught them napping to take the lead four minutes before the break, when Mason raced on to Peter Whittingham's raking pass from inside his own half and held his nerve to lob the ball over the advancing Lonergan from inside the penalty area.
City striker Kenny Miller was then denied in fine style by Lonergan, who kept out the Scot's angled 18-yard drive as Cardiff threatened to increase their lead in time added on at the end of the first half.
Leeds began the second half with renewed purpose. Thompson's fierce shot was blocked by Cardiff defender Ben Turner and Becchio headed wide from 12 yards soon after.
Leeds boss Neil Warnock sent on Ramon Nunez and Danny Webber for McCormack and Adam Clayton respectively just before the hour-mark and the visitors then went close when Danny Pugh's 20-yard effort flew wide.
The visitors continued to give as good as they got, but Lonergan was forced into another excellent save in the 70th minute when keeping out Andrew Taylor's goalbound volley.
Leeds then dealt Cardiff's play-off hopes a blow with the equaliser three minutes later. Substitute Connolly swung over a fine cross from the right and Becchio rose above the home defence to head beyond Cardiff goalkeeper David Marshall into the bottom corner.
Cardiff substitute Filip Kiss fired wide from the edge of the area and Whittingham failed to keep his 87th-minute free-kick on target as Leeds held on for a deserved point.
